WIMP dark matter expected in the parity solution to the strong CP problem

We construct an extended Standard Model (SM), motivated by the strong CP problem and dark matter. In our model, parity symmetry is conserved at high energy by introducing a mirror sector with the extra gauge symmetry, $SU(2)_R \times U(1)_R$. The charges of $SU(2)_R \times U(1)_R$ are assigned to the mirror fields in the same way as in the SM, but the chirality of the mirror fermions is opposite to respect the parity symmetry. The strong CP problem is resolved, since the mirror quarks are also charged under the $SU(3)_c$ in the SM. In the minimal setup, the mirror gauge symmetry leads stable colored particles, so that we introduce two scalars in order to avoid the stable colored particle. Interestingly, one of the scalars becomes stable because of the gauge symmetry, so that it can be a good dark matter candidate. In this paper, we especially study the phenomenology relevant to the dark matter. We also discuss the neutrino sector and show that the right-handed neutrinos from the mirror sector can increase the effective number of neutrinos or dark radiation by $0.14$.


Introduction
The Standard Model (SM) is a very successful model in explaining enormous results at terrestrial laboratories. In particular, the LHC discovered the Higgs boson that was the last piece of the SM. In the meantime, various cosmological and astrophysical observations show that the SM has to be extended in order to account for the dark matter, neutrino oscillation, baryon asymmetry and so on. There are also several theoretical issues in the intrinsic structure of the SM. For example, the SM cannot explain why the mass of Higgs boson is extremely small compared with the Planck scale, namely the gauge hierarchy problem.
Another issue of the SM is the so-called strong-CP problem, which states why θ parameter in the θ-term, is extremely small. The SM consists of quarks, leptons and Higgs field charged under the SU (3) c ×SU (2) L ×U (1) Y gauge symmetry. The parity symmetry is explicitly broken since right-handed fermionic fields are SU (2) L singlets while left-handed ones are doublets. In addition, CP symmetry is also broken by the complex Yukawa matrices. In this framework, the CP-violating θ term is also legitimately allowed so that its values is expected to be O(1). However, the experimental results show that the size of the CP violation is tiny, |θ| 10 −10 [1]. Since there is no way to control θ in the SM, this tininess of θ poses a theoretical problem for the SM known as the strong CP problem. This problem may be a good clue to consider new physics behind the SM. There have been attempts to solve this problem by introducing the Peccei-Quinn symmetry and axion [2][3][4][5], left-right symmetry [6][7][8][9][10], with applications to neutrino physics [11], the baryon asymmetry [12,13], the LHC physics [14], grand unification [15] and flavor physics [16][17][18].
In this paper, we propose an extension of the SM with its mirror sector to respect the parity symmetry so that the strong CP problem is solved. A scalar field acts as a portal between two sectors and this field is a good candidate for the dark matter. In the SM, the parity is explicitly broken, while the symmetry is conserved if we introduce the mirror sector of the SM fields. We extend the gauge symmetry SU (3) c × SU (2) L × U (1) Y to SU (3) c × SU (2) L × U (1) L × SU (2) R × U (1) R and introduce the mirror quarks and leptons charged under the extended gauge symmetry. The θ-term is forbidden because this model respects the parity. The parity is spontaneously broken at some high scale and the SM is effectively generated at the low scale. As discussed later, the loop-induced θ at low energy scale would be negligible even if renormalization group (RG) running is taken into account.
In this setup, candidates for the dark matter are naturally introduced to deplete the lightest mirror quark and lepton. As we shall show in Sec. 2, the mirror quarks and leptons become stable as a result of the gauge symmetry as far as there is no portal coupling between the SM and the mirror sector. If there are some extra scalars as the portal, the mirror fermions decay through this portal coupling. Interestingly, the lightest scalar can be neutral under the SM gauge symmetry and the remnant of the extended gauge symmetry makes the scalar field stable. Thus this scalar field becomes a good candidate for the dark matter (DM).
One of our motivations in this paper is to study the thermal relic density of the scalar field and the constraints from the DM experiments. These DM physics are related to the mirror fermion masses and the couplings between the scalar field and the mirror fermions. This relationship will give upper bounds on the new physics scale because the mirror fermion masses are completely determined by the spontaneous parity breaking scale and there would be upper bounds on the coupling constant to keep the model perturbative up to the parity-breaking scale.
This paper is organized as follows. In Sec. 2, we introduce our model with the parity symmetry and discuss it in detail. Later in Sec. 3, we show how this model solves the strong CP problem. We discuss the DM physics, the flavor physics and the LHC physics in Sec. 4. In Sec. 5, we also study the neutrino sector in the model and its possible cosmological effects. Main results are summarized in Sec. 6. The RG equations in this model are shown in Appendix A and the Higgs portal coupling effect for the DM physics is discussed in Appendix B.

The model with the parity symmetry
In this section, we shall construct an extended model with SU (3) c × SU (2) L × U (1) L × SU (2) R × U (1) R which respects the parity symmetry. The conserved parity symmetry forbids the θ term. We shall also establish our conventions and notations here. In general, the parity transformation of a Dirac fermion field, q i (i = 1, . . . , N F ), is defined as using the parity operator, P , that satisfies P 2 = 1 #1 . When the Lagrangian density L for q i and a scalar Φ is given by L q can be invariant under the parity transformation, depending on the covariant derivatives, D µ and D Φ µ and the field Φ. Let us consider the following covariant derivatives A I µ is a gauge field corresponding to a gauge symmetry labeled by I (I = U (1), SU (2)). g I and Q I q,Φ denote the gauge coupling constant and charges respectively. Here, the gauge field and scalar fields are defined as #1 We have adopted the convention P = P −1 . so that A I µ and Φ are the 4 × 4 matrices that do not commute with γ µ and each of the elements is linear to the unit matrix of 2 dimension. Note that the fermion q i is decomposed into the right-handed and the left-handed components, i.e. q i = (q i R , q i L ) T , in this description. A I µ R and A I µ L correspond to the gauge fields for the gauge symmetry that act on q i R and q i L , respectively. The scalar field Φ and the gauge field are transformed under the parity as

Fields spin
Thus, the parity transformation leads the following exchange: The Lagrangian L q becomes invariant under the parity transformation as far as the γ 0 dependence does not show up explicitly in V (Φ). The gauge interaction should also respect the parity symmetry: where F I µν is the field strength composed by A I µ . One can immediately see that the θterm is not allowed by parity symmetry. Note that there are two chiral gauge symmetries described by A I µ L and A I µ R that have the same gauge coupling. In addition, we can find the gauge kinetic term in the U(1) gauge symmetry case.
Based on this generic argument, we extend the SM to the parity conserving model. In the SM, the gauge symmetry is G SM = SU (3) c × SU (2) L × U (1) Y . Now, we extend the gauge symmetry as Fields spin In the SM, the charge assignment of SU (3) c is vector-like, so that we only consider the diagonal direction of SU (3) L × SU (3) R , that is identified to SU (3) c . Based on the argument above, the parity transformation leads the exchange of the symmetry: This extension has been proposed to solve the strong CP problem [7] which however does not have a dark matter candidate. The matter content in the original side is summarized in Table 1.
where H L is defined as H L = iσ 2 H * L . σ 2 is the Pauli matrix. We introduce a mirror sector to respect the parity symmetry as follows. The matter content of the mirror sector is summarized in Table 2. Q i L , u i R , and d i R (i = 1, 2, 3) are the mirror quarks charged under SU (3) c × SU (2) R × U (1) R . The fields, l i L and e i R , denote the mirror leptons. H R is a scalar charged under SU (2) R but not under SU (2) L . The vacuum expectation value (VEV) plays a role in making the mass hierarchy between the original side and the mirror side. The detail will be shown below.
The Yukawa couplings among the mirror fields are written down as follows: Note that the Yukawa couplings are defined to respect the parity symmetry, that corresponds to the following exchange: The structure of the mirror sector is the same as the one of the original side, because of the parity symmetry. Then, we expect that some stable particles appear in the mirror sector in the same way as the proton and electron in the SM. Those stable particles are, however, strongly constrained by the cosmological observations and the searches for the stable extra charged particles. Below, we study the stability of the mirror particles and investigate the possibility that some neutral particles become a candidate for the cold DM. Then, we propose one extension to avoid the stable charged particles.

Stability of the extra particles and dark matter candidate
The VEV of H L breaks down the EW symmetry to the electromagnetic (EM) symmetry, U (1) em . Let us consider the case that the EM charge of the field q, Q q em , is given by where τ q L is the isospin given by the third component of SU (2) L and Q q L is the charge of U (1) L . In this case, the mirror particles are not charged under U (1) em . In our model, the non-vanishing VEV of H R breaks SU (2) R × U (1) R down to the mirror U (1) m em , that is different from the EM symmetry in the SM.
We find that the mirror quarks cannot decay to the SM quarks in this scenario. For convenience, let us define the subgroup of U (1) em : U (1) em ⊃ Z em 3 . In the SM, the up-type quarks u i and down-type quarks d i are charged under SU (3) c × Z em 3 as follows: where ω 3 = 1 is satisfied in our notation. We note that the other SU ( gauge symmetry to break the U (1) R em symmetry spontaneously, the remnant symmetry from U (1) R em would guarantee the stability of the U (1) R em -charged particles. Such stable mirror particles may lead to an unfavorable consequence. At the QCD (de)confinement transition in the early universe, the mirror up quark would form stable exotic hadrons together with the SM light quarks, such as u ū and u uu. Since these hadrons are fractionally charged and scatter with visible matter via strong or EW interaction, the cosmological abundance is strongly constrained. For stable colored particles much heavier than the confining scale, the abundance of the exotic hadrons has been estimated in the literature [19][20][21], taking non-perturbative effects at or below the QCD scale into account, as where m and Λ QCD denote the colored particle mass and the QCD confinement scale, respectively. This gives a small value of O(10 −4 ) for m = O(TeV), while the direct searches for strongly interacting particles and fractionally charged particles will put severe constraints on their flux at the Earth surface [22]. Note that precise prediction for the cosmological abundance and the experimental bounds require a full knowledge of the non-perturbative QCD. Thus, further careful study is needed to conclude viability of this scenario, and it will be pursued elsewhere.
In addition, the mirror electron is also stable in this case. The thermal abundance set by e ē → γ γ is estimated as Ω e h 2 0.1(m e /100 GeV) 2 . The mirror electron mass is correlated with the mirror up quark mass, since their masses are given by the Yukawa coupling constants which are fixed by the SM Yukawa coupling constants at the parity breaking scale. The LHC limit on the mirror up quark comes from the search for the so-called R-hadron which is composite states involving supersymmetric particles. The lower limit on a top squark mass is about 890 GeV [23] from the search for the R-hadrons coming from top squark pair production. The limit on the mirror up quark in this model is estimated as 1080 GeV by assuming the pair production cross section of the mirror up quark is four times as that of top squarks. This means that the mirror electron should be heavier than 250 GeV, and the relic would then be too abundant. Further, if there is gauge kinetic mixing between U (1) L and U (1) R , the mirror electron can be millicharged. The stable millicharged particle can affect the CMB power spectrum, and hence, for 2 5 × 10 −9 (m e /100 GeV), the abundance should satisfy Ω e h 2 10 −3 [24]. This can be another constraint on this scenario.
In this paper, in order to avoid the stable colored particles and overproduced mirror electron, let us consider another case where Q q em is given by where Q R q em is the charge of U (1) R em , τ q R is the isospin given by the third component of SU (2) R and Q q R is the charge of U (1) R . This breaking pattern is realized in a situation by introducing extra scalars charged under both U (1) L and U (1) R . We introduce such Fields spin scalars denoted by X b and X l that are singlet under SU (3) c × SU (2) L × SU (2) R , but have charges under both U (1) L and U (1) R as defined in Table 3.
In our study, we consider two cases: (I) X b = 0 and X l = 0, (II) X b = 0 and X l = 0.
Note that H R also develops nonzero VEV in both cases. The U (1) R em gauge symmetry is broken by either X b or X l , and a subgroup of U (1) R em remains unbroken similarly to Z em 3 discussed before. In the case (I), the unbroken symmetry is Z R 3 , while the unbroken symmetry is Z R 2 in the case (II). The scalar, , so that it is stable as far as its VEV is vanishing. We note that the scalars are neutral under the EM symmetry according to Eq. (18) and the charge assignment in Table 3. #3

The interaction of the scalars
We consider interactions between the additional scalars and the fermions. The scalars, X b and X l , are only charged under U (1) R × U (1) L as shown in Table 3. The charge assignment is defined to make the extra quarks and leptons unstable. In this setup, the Yukawa couplings are written as: The parity symmetry forces λ ij u and λ ij e to satisfy λ ij The extra fermions can decay to the SM fermions and X b,l through these Yukawa couplings.
Next, let us discuss the gauge interaction and the scalar potential. The Lagrangian involving X b,l is given by #3 If the (U (1) L , U (1) R ) charge of X b is defined as (−1/3, 1/3), X b couples to down-type quarks at the renormalizable level and the phenomenology is similar to the one discussed in Refs. [25,26]. In this case, however, we may suffer from the bound on the stable mirror up quark.
where (Q b , Q l ) = (2/3, −1) is defined. A L µ and A R µ are the gauge fields of U (1) L and U (1) R symmetries, respectively. V S is the potential for the scalar fields: The scalar potential V S induces the symmetry breaking, depending on the mass parameters in V S . We note that all parameters in V S can be defined as real valued, so that there is no contribution to the θ term. Conditions for a certain symmetry breaking is discussed in next subsection. Before the study of the gauge symmetry breaking, we see the U (1) L × U (1) R gauge kinetic terms. The kinetic terms of the gauge fields are where is the kinetic mixing allowed by the parity symmetry. To summarize, the parity transformation exchanges the gauge fields and the scalar fields as

The condition for the gauge symmetry breaking
We study the vacuum structure given by V S and find out the condition for the parity and gauge symmetry breaking. We expect that the VEVs of the scalars are not vanishing and each of them causes the following symmetry breaking: We discuss the symmetry breaking one by one. The scalar fields have the VEVs as As mentioned above, the VEVs are real since the parameters in V S are real. The stationary condition for H R gives the equation for the VEVs: The VEV of X b (X l ) is vanishing, in the case (I) (case (II)). Let us focus on the case (I). Note that the results can be applied to the case (II) by replacing the index b with l. The stationary condition for X l is described aŝ In our setup, v L breaks the EW symmetry and is assumed to be tiny compared to the other VEVs. Then, assuming v L v R , v l X , the stationary conditions, Eqs. (30) and (31), lead an approximate condition for v R and v l X : whereλ =λ + +λ − is defined. When we choose the appropriate parameters in the righthand side, we can realize the symmetry breaking in Eqs. (26) and (27). Next, we discuss the EW symmetry breaking. The Higgs boson H L should develop the non-vanishing VEV to cause the EW symmetry breaking. After the parity symmetry spontaneously breaks down, the effective scalar potential can be evaluated as The parameters in the effective potential are renormalized, taking into account the corrections from v l X and v R . In particular, m 2 ef f is approximately evaluated as at the tree level. m 2 ef f is expected to be the source of the EW symmetry breaking, so that λ − should be tiny to obtain the EW symmetry breaking scale that is much smaller than v R . Ifλ − is vanishing, the global symmetry in V S is enhanced, so that the direction of H L could be interpreted as the pseudo-Goldstone boson. In such a case, we may obtain the EW symmetry breaking radiatively [27].
In our work, we focus on the phenomenology without the analysis of the radiative correction to the scalar potential. We simply introduce a soft parity breaking term for H L : Thus, the EW symmetry breaking scale is realized although we may have to allow the fine-tuning for the Higgs mass term. This setup can evade the domain wall that would be generated by spontaneous parity breaking. In the case (I), the VEV of X b is vanishing and the mass is given by m 2 b . We note that the mass m 2 b depends on v R and v l X through the quartic couplings, namelyλ bl andλ b H . Then, the mass of X b is expected to be around v R and/or v l X . As will be shown in Sec. 4, the VEV v R needs to be much higher than the EW scale to avoid the experimental constraints. X b may reside at the very high-energy scale ∼ v R . The mass scale of X b , however, depends on the parameters in V S , so that we study phenomenology without specifying the underlying theory of our model and treat m 2 b as a free parameter.

The solution to the strong CP problem
We discuss the strong CP problem in this section. In general, the θ parameter is described effectively as where θ is from the non-perturbative effect of QCD vacuum and m u,d are the mass matrices for the SM quarks. The upper bound on θ from the experiment is about 10 −10 [1]. In our model, the parity symmetry is respected, so that the θ term is forbidden. Note that the θ term explicitly breaks not only the CP symmetry but also the parity symmetry. This kind of scenario has been proposed motivated by the strong CP problem [6,[8][9][10]13]. The parity symmetry is respected by introducing the mirror sector at some high-scale until it is broken spontaneously. This means that the parity is broken to some extent at low-energies, e.g. the EW scale. An important fact of this model is that the Yukawa matrices for the mirror fermions are same as the SM ones at the high scale where the parity is conserved. The parity breaking scale, or equivalently the extra gauge symmetry breaking scale, should be higher than about 10 8 GeV to make the first generation mirror fermions heavier than current experimental limits. Then, the RG correction to the θ term may be non-negligible as well as the corrections from the threshold and the higherdimensional operators.
First, let us discuss how the θ term is vanishing near the parity breaking scale. In our model, the parity symmetry forbids the θ-term at tree-level, namely θ = 0. Due to the existence of mirror particles, the quark matrices, m u and m d , are replaced by where A u,d and B u,d are 3 × 3 matrices. At the renormalizable level, A u,d and B u,d are vanishing in our model. Then, one can immediately realize which is a real number. Therefore, the θ parameter, given by , is vanishing. The symmetry breaking may effectively generate nonzero A u,d and B u,d . In the case (I), X b does not develop the VEV and the remnant symmetry is Z R 3 #4 . Since the mirror quarks are charged under Z R 3 , the mass mixing terms, A u,d and B u,d are forbidden. Thus, the θ term is not generated even at the low energy. We give a discussion about the loop correction later. #4 Actually, in this case it is an accidental global U (1) symmetry which preserves mirror baryon number.
In the case (II), the VEV of X b is not vanishing, while that of X l is vanishing. The remnant symmetry is Z R 2 . The mirror leptons and the mirror down-type quarks are Z R 2 -odd while the mirror up-type quarks are Z R 2 -even. This causes that A d and B d are forbidden, but A u and B u are not. In fact, the VEVs of the scalars generate where Λ is a cut-off scale. If Λ is very large, A u is vanishing and θ is also vanishing. Otherwise, the Yukawa couplings, λ ij u and a ij u , should be suppressed to evade the bound from the CP violation, in the case (II) with v b X = 0. Next, let us discuss the RG corrections. The relation in Eq. (39) is modified by the RG corrections and this could revive the strong CP problem at low-energies. The RG equations for the determinants of Y d and Y u are given by where β Y d and β Yu are the β-functions defined as If the right-hand sides of Eqs. (41) and (42) are complex numbers and the imaginary parts of the determinants are amplified, the sizable θ term is predicted at low scales. We note that the imaginary parts of det(Y d ) and det(Y u ) can be set to zero at the initial condition, according to the phase rotation of quarks and the mirror quarks. Let us discuss the beta functions explicitly at the one-loop level. After H R develops non-vanishing VEV, we could integrate out the mirror fermions. Then, the beta functions, β Y d and β Yu , are evaluated as where γ H is given by This leads the RG equations for det(Y u,d ) as Thus, the imaginary parts of the left-handed sides of the RG equations are not evolved, since the right-handed sides are real at the one-loop level. In Appendix A, the relevant RG equations are summarized.
With the same spirit as in Ref. [28], the contributions from renormalization of quark mass matrices alone are around O(10 −16 ) even at the higher loops. The main differences in our model are the introduction of X b , possible mixings in the Higgs sector and the kinetic terms of U (1) symmetries. These terms might lead to non-vanishing corrections at loop levels. In our model, the structure of chirality, the parity symmetry and the heavy masses of the mirror quarks however suppress the loop correction. Following Ref. [28], the three-loop diagrams involving λ u coupling would lead non-vanishing θ term in the case (I), while the one loop diagram involving the CP-even scalars would contribute to θ in the case (II) because of the non-vanishing VEV of X b . In both cases, their contributions are suppressed by λ u so that the loop correction would not spoil the tininess ofθ as far as the size of λ u is not too large. In fact, we assume that the alignment of λ u is unique to avoid the flavor constraints. Then, the loop correction to θ is much suppressed.

Phenomenology
We study the phenomenology in this section. In our model, there are Yukawa couplings involving scalars, the mirror fermions and the SM fermions: as shown in Eq. (20). Because of the parity symmetry, λ ij ψ is a hermitian matrix and the mirror fermion mass ratios are the same as the SM predictions above the parity breaking scale. The mirror up quark and electron are the lightest mirror quark and lepton that are expected to dominantly contribute to the low-energy physics. Note that λ ij ψ is defined in the mass base. Our main motivation of this paper is to study physics involving our DM candidate. In particular, we will numerically analyze the parameter region allowed by the DM physics and discuss the flavor and LHC physics relevant to the result. We study the phenomenology of the case (I) in Subsection 4.1 and of the case (II) in Subsection 4.2.

Case (I) : baryonic DM (X b ) scenario
We consider the case that X b does not develop the VEV. In this case, the Z R 3 symmetry remains unbroken as the remnant of the subgroup of U (1) R em and this makes X b stable. The scalar X b couples to the SM up-type quarks and the mirror quarks via the λ u coupling. Since the scalar X b is stable and couples with the SM particles involving the mirror up quark which may lead a suitable co-annihilation cross section, it is a good DM candidate. We assume that only the Yukawa interactions involving the mirror up quark are relevant to phenomenology. Furthermore, we consider the following three cases that X b dominantly couples to . Below we discuss the predictions and constraints of each case. Note that contours for these parameters in each case should be interpreted as upper bounds since in principle these couplings could be present simultaneously. In the Higgs-portal DM scenario, there are many discussions in the literature [29][30][31][32][33][34], so we shall not repeat here. In Appendix B, the contribution of the Higgs portal coupling between DM and the SM Higgs is summarized and the impact on our result is shortly discussed.

Dark Matter Physics
To begin with, we discuss DM relic abundance, direct detection and indirect detection in the X b DM scenario, assuming that X b was thermally produced in our universe. These observations put constraints on the Yukawa couplings, λ ij u and masses of the DM and the mirror fermion. We first study general features in this kind of model, and then elaborate on each case listed above.
This DM candidate mainly annihilates into a pair of the up-type quarks by exchanging the mirror quarks in the t-channel. Now we assume that flavor violating annihilations, , are negligibly small, and then the dominant processes are flavor conserving annihilations. In the non-relativistic region where the thermal freeze-out occurs, the cross section can be expanded in terms of the relative velocity of incoming DM particles, v, with in the massless quark limit (m j , m X m i ). m X denotes DM mass and m i and m j masses of the SM and mirror up-type quarks: The partial s-wave of this process is suppressed by the quark mass in the final state. Thus, the pair annihilation will be p-wave dominant except for the top-philic case, (C). In the other cases (A) and (B), the large Yukawa couplings are required to achieve the observed relic abundance.
The Yukawa couplings, λ ij u , also give rise to elastic DM-nuclei scattering. The DM-quarks effective interaction relevant for the spin-independent (SI) scattering is given by At tree level, the mirror fermion exchanging, as shown in the left panel of Fig. 1, generates C S,q , C T,q and C V,q : where we neglect the SM quark masses. DM particles can also scatter off the gluon in the nucleon via box diagrams in the central panel of Fig. 1. Integrating out the short-distance contribution, we find the coefficient to be Note that this equation is valid only when the mirror quarks and DM are sufficiently heavier than the SM quarks. For the result which keeps the quark masses finite, see e.g. Ref. [35].
There are important loop processes as well. At one-loop level, photon and Z boson can mediate the DM-nuclei scattering via penguin diagrams as shown in Fig. 1. The photon exchanging induces the DM coupling to the quark vector current with the coefficient, where with Similarly, the contribution from the Z boson exchanging is evaluated as where g V,q = (T 3 ) q − 2Q q sin 2 θ W and In the limit that m j m i , m X , Eqs. (59) and (62) reduce to simpler forms, We find that the Z-exchanging contribution is proportional to the quark mass squared, and then it will be significant only in the top-philic case. Below, we only keep contributions from the mirror up quark, and discuss the cases (A), (B) and (C) where the mirror up quark dominantly couples to the up, charm and top quark, respectively. The DM candidate is simply denoted as X.

(A) Up-philic case
In this case, the DM particle X scatters off the up quark, that is a valence quark in nucleons, at the tree level. The up-philic coupling is strongly constrained from direct detection experiments.  Let us estimate the elastic DM-nucleon scattering cross section, assuming the DM abundance was produced via this coupling. From Eq. (51), the DM pair annihilation cross section is given by with good approximation. This formula shows how λ uu u , m X and m u are related to each other via the observed abundance. Using an approximate solution for the thermal relic abundance, Ωh 2 0.12 × 10 −36 cm 2 / σv , the SI scattering cross section is approximately given by This is several orders of magnitude larger than the current XENON1T bound [36], thus we conclude that the up-philic case has already been excluded.

(B) Charm-philic case
Charm-philic case is similar to the up-philic one in the DM annihilation, so a large Yukawa coupling is required for the relic abundance. However, since the charm quark is a sea quark in nucleons, the tree process does not generate the couplings of DM to nucleon vector current and thus the SI cross section is much smaller than the up-philic case. In this case, the dominant contribution comes from the one-loop photon exchanging in most  parameter space. The exception is a compressed region, m X m u , where DM-gluon scattering via box diagrams can dominate over the former contribution.
In Fig. 2, we show how various contours in the parameter space are confronted with relic abundance, direct detection and perturbativity. The yellow area gives Ω X < Ω CDM where coannihilation processes are too efficient and the produced DM abundance is below the observed one even for λ cu u = 0 #5 . The pink regions are excluded by LUX (solid) [37] and XENON1T (dashed) [36] experiments. On the left-panel, regions above the green contour labeled with Landau pole shall give too big coupling below the W R boson mass scale, part of which has been already constrained by LUX and XENON1T. The detail of our analysis on this bound is shown in Appendix A. The gray region has λ cu u > √ 4π, a strongly-interacting theory already at TeV scale. As a result, only white region in the right panel can satisfy the observed DM abundance and escape various constraints. The blue dashed lines on the right panel show the mirror electron mass translated from the mirror up quark mass using m e (m e /m u ) m u .

(C) Top-philic case
The top-philic case is more complicated than the other two cases. Since the top quark is much heavier than the other quarks, the s-wave contribution is not so suppressed in XX † → tt. Then, a smaller Yukawa coupling, λ tu u , is predicted in this case. In direct detections, the Z exchanging can make a dominant contribution, instead of photon exchange process, because of the large top mass. Fig. 3 shows various contours and constrains, similar to the charm-philic case. We can see that the white region on the right #5 In fact, λ cu u cannot be vanishing and has to be large enough for Xg ↔ u ū i process to frequently occur at freeze-out. In our case, the condition is fulfilled for λ cu panel is a bit larger than the charm-philic case. Note that DM models with a top partner has also been studied in other literatures, e.g, in Refs. [38,39].
We point out that in all the above cases indirect searches from cosmic rays, gamma-ray and neutrinos do not pose any pressing limit, due to the p-wave suppression. In the early universe when DM was freezing out, the velocity is about 1/3, while at the present time v ∼ 10 −3 . Therefore the annihilation cross section at present is 10 −6 smaller than the canonical value for the thermal relic.

Flavor Physics
In our study, we assume that only one element of λ ij u is sizable and the others are negligibly small. Phenomenologically, the processes involving the lightest mirror quark, u , are the most sensitive to the physical observables at low energy. Then, the relevant Yukawa couplings between the mass eigenstates of the fermions and X b are In the DM physics, we investigate the three cases: (A) up-philic case, (B) charm-philic case, and (C) top-philic case. In general, constraints from flavor physics are very tight, even though the new physics scale is much higher. In this subsection, we discuss the constraint from flavor physics relevant to the DM physics in each case, taking into account the small Yukawa couplings irrelevant to the DM physics as well.
In the baryonic DM scenario, namely case (I), the DM candidate X b interacts with the up-type quarks via the Yukawa couplings. There are also gauge interactions induced by the Z and the Z-Z mixing, but they are suppressed by the large Z mass. The dominant contribution to flavor observables is effectively induced by the terms in Eq. (68).
First, let us discuss the cases (A) and (B). In those setups, either |λ uu u | or |λ cu u | is large. If the other element is also sizable, flavor violating couplings would be effectively generated by integrating out the mirror quark and X b . For instance, if |λ cu u | (|λ uu u |) is not vanishing in the case (A) (in the case (B)), the four-fermi coupling that contributes to the D-D mixing is generated at one-loop level. Such a ∆F = 2 process is generally the most sensitive to new physics, so that we numerically estimate the bound on λ ij u below. The effective operator that contributes to the D-D mixing is generated by the box diagram involving the mirror quarks and X b : where C D is evaluated at one-loop level as x i is defined as x i ≡ m 2 i /m 2 X and f (x) is the function satisfying In the cases (A) and (B), C D is approximately estimated as assuming m u ≈ m X . One relevant observable concerned with the D-D mixing is the mass difference described as The parameters on the right side are numerically known as m D = 1864.83 ± 0.05 MeV, f D = 212.15 ± 1.45 MeV, η D = 0.772, andB D = 0.75 ± 0.02 [40,41]. This value is measured by the experiment, and should be small to evade the conflict with the experimental results. In Ref. [41], the measured value of ∆M is 0.04 % ≤ ∆M τ D ≤ 0.62 % at 95 % CL. If we require the new physics contribution to be less than 0.1 %, we can obtain the bound on λ ij u in the cases (A) and (B) as λ uu u λ cu u 0.005 (0.01), when m u ≈ m X is imposed and m X is fixed at 500 GeV (1 TeV).
In the case (C), on the other hand, the strong bound from the meson mixing becomes much milder, since u dominantly couples to top quark. The exotic top decay in association with a gauge boson in the final state may severely constrain our model instead. The current experimental upper bound on such a process is shown in Ref. [40], and they are roughly O(10 −4 ). For instance, the flavor-violating top decay to photon or gluon and one light quark is induced by the following operator generated at one-loop level: where σ µν = i 2 [γ µ , γ ν ] is defined. F µν and G a µν are the gauge field strengths that consist of photon and gluon, respectively. C ij u is given by where f 7 (x) is defined as Using the coefficients, each partial decay width of the flavor-violating top decays can be estimated. In particular, the decay to a light quark and gluon is larger than the other, because of the relatively large gauge coupling. We conclude that our prediction of the branching ratio is less than 10 −6 and safe for the current experimental bound, even if the Yukawa couplings are O(1) and the DM mass is O(100) GeV. In our model, the flavor-violating top decay associated with a Z boson is also possible, but the prediction is also much below the current experimental bound. Eventually, the strongest bound on our model in the case (C) comes from the direct search for the mirror quarks at the LHC. Figure 4: Typical Feynman diagrams that produce mirror fermions f which subsequently decay into SM fermion f and dark matter X or SM gauge boson V .

The LHC physics
At colliders, mirror fermions could be produced if they are light enough. For example, mirror quarks can be pair produced at the LHC. Typical Feynman diagrams are shown in Fig. 4. The produced mirror fermions decay into a SM fermion, together with dark matter X and/or SM gauge bosons. In case (I) where X l gets nonzero VEV, the mirror electron decays as e → l + Z/W/h induced by the mixing with SM leptons. The Yukawa coupling λ u induces a decay of the mirror up quark: The mirror electron is expected to be the lightest mirror fermion as the SM fermions. A type of a daughter lepton depends on the Yukawa couplings λ ie e and that of a daughter boson depends on how it mixes with the SM leptons. There are studies about the limits on such extra leptons decaying to a lepton and a SM boson [42][43][44][45][46]. A conservative limit may be obtained by assuming that the daughter lepton is exclusively tau lepton. In Ref. [46], it is shown that the limit is at most 250 GeV with an integrated luminosity about 100 fb −1 at √ s = 13 TeV. The limit becomes the tightest if a mirror electron exclusively decays to a Z-boson (and a tau lepton), while there is no limit with the integrated luminosity of 100 fb −1 if the mirror electron decays to all of the bosons with a certain branching fraction. Therefore the mirror electron above 200 GeV could be allowed by the current data at the LHC, although the detail depends on parameters which do not have significant correlations with the DM and flavor physics discussed above.
The mirror up quark should be degenerate with the DM particle in order to explain the relic density, and it should dominantly couples to a charm quark (case B) or a top quark (case C). In case (C), since the mass difference is smaller than the W-boson mass, the mirror up quark could decay via four-body decay process, where W * , t * are off-shell W-boson and top quark, respectively. f 1,2 are the SM fermions coming from W * . The partial decay width may be so suppressed that two-body decay u → c + X b dominates the mirror up quark decay even if the Yukawa couplings possess the hierarchy λ cu u λ tu u . Thus the mirror up quark is expected to dominantly decay into a charm quark and a singlet scalar in both of the case (B) and (C). The current limit on pair produced top squark decaying to a charm quarks and the lightest (neutral) supersymmetric particle is about 500 GeV when the mass difference between a top squark and an invisible particle is larger than 40 GeV [47,48]. The cross section of the mirror up quark pair production is roughly four times larger than that of the top squark pair production, and a top squark with about 630 GeV gives quarter of a pair production cross section of top squark with 500 GeV [49]. Therefore the current limit on the mirror up quark is estimated at about 600 GeV. In case (A), although this case cannot explain the relic density, the mirror up quark decays as u → u + X b . The signature at the LHC is similar to a squark decaying into an invisible particle and a SM quark which give signals with two jets and missing energy. In the mass degenerate region, limits on the squark mass is about 650 GeV [50,51], under an assumption that light-flavor squarks have a common mass. The cross section of the up quark pair production is about a half of that of the squarks, so that the limit is estimated as about 600 GeV [49]. Note that this search is also relevant for case (B) and (C), because their difference is if the c-tagging is exploited or not. Thus, in any case, we expect that the LHC limits on mirror up quark is about 600 GeV in the mass degenerate region.
The mirror down quark is very long-lived in our model, because it does not couple to the SM particles through the Yukawa couplings in contrast to the mirror up quark and electron. The mirror down quark decays only through the W R -boson exchange, so that the decay rate is suppressed by the parity breaking scale ∼ v R . The decay width is estimated as where the RG corrections to the gauge and Yukawa couplings are neglected. The decay width is about three orders of magnitude smaller than the decay width of the muon. The mirror down quark is expected to be hadronized before it decays, and pass through the detector of the LHC. This kind of signal is studied in analyses searching for the so-called R-hadrons, which is composite colorless states involving supersymmetric particles [23,52]. The result [23] gives limits on the mass of bottom squark about 800 GeV based on a model that the R-hadrons are originated from the bottom squark production. Since the pair production cross section of the mirror down quark is expected to be twice times as that of the bottom squark if they have same masses, the limit for the mirror down quark is estimated as about 890 GeV. The mirror down quark mass is about eight times heavier than the mirror electron mass as expected from the SM fermion masses. The current limit may be satisfied even if the mirror electron is about 200 GeV and the mirror down quark is about 1.6 TeV. This signal is an interesting possibility to be discovered in the future results at the LHC.

Case (II) : leptonic DM (X l ) scenario
Similarly, we can discuss the leptonic DM case where X b develops non-vanishing VEV and X l does not. In this case, the Z R 2 symmetry remains as the remnant of the subgroup of U (1) R em and this makes X l stable. Then, X l is a cold DM that couples to the charged leptons via the λ e couplings. In the same manner as Sec. 4.1, we study the DM physics, assuming only the mirror electron makes sizable effects on DM phenomenology because of its light mass, and one of the λ e couplings dominates over the others, e.g., |λ τ e e | |λ µe e |, |λ ee e |.

Dark Matter Physics
The DM physics in the lepton-philic scenario can be understood directly from analysis in the case (I). The annihilation is p-wave dominant due to the light charged lepton masses, so that Yukawa couplings have to be large enough to account for the DM abundance. Direct direction is simple as well. The DM-nuclei scattering is caused only through photon and Z exchanging, because the DM particle does not directly couple to any colored particles. Besides, the light lepton masses lead the negligible Z-exchanging contribution. Thus, the main process is photon exchanging in the whole parameter space. We would like to note, however, that we need a modification in Eq. (59) in the electronphilic case. This equation was derived assuming transfer momentum is negligibly small compared to particle masses in the loop. The typical transfer momentum is ∼ 50 MeV for a xenon detector, for example. Therefore, Eq. (59) is invalid in the electron-philic case, and we have to modify the expression by taking a finite momentum transfer into account. Figure 5 shows how parameter space is constrained in the tau-philic case in the same manner as in the charm-philic case. There is no big difference between the muon-philic and the tau-philic case. The electron-philic case is strongly constrained by the EW precision measurement, given by the LEP experiment. In the white region, the relic density is explained without conflicting to all the constraints, but it is very narrow.

Flavor Physics
In the leptonic DM case, we assume that one element of λ ij e is sizable. In such a case, one of the stringent constraints comes from l 2 → l 1 γ. The processes are given by the dipole operators: C ij e is estimated at one-loop level following the result on the exotic top decay in Sec. 4.1.2: In particular, the flavor-violating muon decay, i.e. µ → e γ, is severely constrained by the experiment: Br(µ → e γ) < 4.2 × 10 −13 [53]. In the muon-philic DM or electron-philic DM case, the exotic decay may be enhanced by the Yukawa coupling. The upper bound on λ ij e is estimated as λ ee e λ µe e 0.002 (0.009), when m e ≈ m X is imposed and m X is fixed at 500 GeV (1 TeV). The flavor-violating τ decays, i.e. τ → e γ, µ γ are less constrained and we confirm that our predictions are below the current bounds which is O(10 −8 ) [40] even if the Yukawa couplings are O(1).
As another process, the lepton flavor violating (LFV) process, l j → l i l k l k may become sizable depending on setups. The contribution to the process is given by two types of diagram: the box diagram and the penguin diagram. The box diagram is much suppressed in our setup, since it is linear to one sizable λ je e and suppressed three couplings, λ ie e λ ke e λ k e e , for instance. If l k is identical to l k or l i , the penguin diagram is possible. We can estimate the upper bound on the Yukawa coupling, using the experimental upper bound on the muon decay, µ → 3e as [54,55] λ ee e λ µe e 0.06 (0.23), when m e ≈ m X is imposed and m X is fixed at 500 GeV (1 TeV). Thus, we conclude that the bound from µ → eγ is more important in our model. Similarly, we can discuss the LFV decays of τ , but our prediction is much below the experimental bound, even if the Yukawa couplings are assumed to be O(1). We have also estimated our prediction on the µ-e conversion process in nuclei, but the bound is also not stronger than the one from µ → eγ.

The LHC physics
The relevant processes for the mirror fermions are also given by Fig. 4 as in Case (I), while the mirror electron decays as e → l + X l and the mirror up quark decays as u → u + V . The signal of the mirror down quark is not changed from the case (I).
In this case, the mirror electron pair production gives the same signal as the slepton pair production. In parameter space where a mass difference between a slepton and an invisible particle is less than 10 GeV, the lower limit is at most 190 GeV under an assumption that all of the selectron and smuon have a common mass [56]. This limit is expected to be directly applicable for the mirror electron, because the production cross section is similar to the sleptons pair production in the analysis [56]. There may be no limits in the moderate mass difference region. If the mass difference is about 50 GeV (200 GeV), the lower limit is about 200 (500) GeV under an assumption the mass difference with degenerate selectron, smuon and stau [57,58]. The limits will be slightly weaker for the mirror electron in this model due to the smaller production cross section. If the mirror electron exclusively decays to a tau lepton and a DM particle, the signal will be too small to give bounds on the mirror electron mass [59].
The signature of the mirror quark decay u → q + Z/W/h will be similar to the vectorlike quark searches [60,61]. As for the e → l + Z/W/h in case (I), the limit depends on flavor of a daughter quark and a type of daughter boson. The limits on vector-like partners of the third-generation quarks are studied in Ref. [60], and a limit on top (bottom) quark partner is 1.31 (1.03) TeV for any combination of decay mode. In this section, we discuss the neutrino sector. Since neutrino oscillation experiments indicate that at least two of three SM neutrinos has to be massive, we shall be able to accommodate massive neutrinos in this model. To accomplish that, we introduce fermionic fields N i that are neutral under the gauge symmetry as shown in Table 4. N i are transformed by the parity as

Neutrino sector
Then, the Yukawa couplings concerned with the neutrino masses are given by In addition, we can write down all the possible mass terms: The first and second terms correspond to the Dirac and Majorana mass terms, respectively. Depending on the size of each mass term, we can discuss some possibilities. If M ij vanishes, the active neutrinos are Dirac fermions. If m ij vanishes, the active neutrinos are Majorana fermions. We consider both cases below.

Dirac neutrino scenario
If N i is expected to be charged under the U(1) lepton symmetry, the Majorana mass matrix, M ij , is forbidden. Assuming |m ij | v R , we obtain the tiny Dirac mass matrix for the active neutrinos: Thus, the active neutrino mass matrix, m ν , is evaluated as Naively, introduction of so many extra neutrino states would be in conflict with current cosmological bounds on neutrino masses and number of light species. However, we can actually show that all those new components were never produced abundantly in the early Universe if the relevant couplings are small or new mass scales are high. In the Dirac neutrino case, the three right-handed neutrinos ν R could be abundantly produced due to its SU (2) R gauge interaction if the universe was hot enough, but they will decouple earlier. They will contribute to effective number of neutrinos by where T ν L is the three active left-handed neutrino's temperature, equal to (4/11) 1/3 T γ after e ± 's annihilation, T dec for the temperature at which ν R decouples from the SM sector, g * s counts the effective number of degrees of freedom for entropy density in the SM. We can get a lower bound on δN eff before e ± 's annihilation when the decoupling temperature T dec is larger than top quark's mass, This number is well below the Planck's limit, δN eff 0.30 [62]. Since the ν R and ν L combine into a Dirac neutrino, they would have the same mass. Therefore, current cosmological limit on neutrino's mass is also safe in this scenario. Future experiments would have the sensitivity to probe δN eff ∼ 0.02 − 0.03 [63].

Majorana neutrino scenario
If the lepton symmetry is not assigned, the Majorana mass matrix is allowed and the tiny neutrino masses can be generated by integrating out the heavy N i : in the limit that the Dirac mass matrix, m ij , is vanishing. The first term gives the active neutrino masses. Cosmological constraint in this case becomes more complicated than in Dirac case. We have two copies of seesaw mechanism. Then for each generation, after mass diagonalization we would have one active Majorana neutrino and three new Majorana ones. Two of the three new neutrinos can be very heavy due to the Majorana mass M ij and they can decay quickly. The remaining one has a mass m ν × v 2 R /v 2 L where 0 ≤ m ν 0.1 eV. Since we expect v R /v L > 10 6 at least, we would have a O(100) GeV neutrino for m ν ∼ 0.1 eV. If abundant and stable, they will contribute too much to the energy density and overclose our Universe. Fortunately, one of the three active neutrinos can be massless, which also means one of the three mirror neutrinos can be massless. Since neutrino mixing occurs also in mirror sector, we would have the decay process for heavy mirror neutrinos (ν H ) into three massless mirror neutrinos (ν 0 ) through the Z R mediator, ν H → 3ν 0 , with decay width (92) As long as Γ 1 s −1 , the heavy mirror neutrinos will decay before BBN era. However, ν H should decay where it is still relativistic, otherwise the decay products would constitute too much dark radiation. This put a constraint on Γ 10 10 s −1 v R /v L 10 6 4 m ν 0.1eV 2 . (93) Combined with Eq.(92), it would give v R /v L > 10 11 , a very stringent limit. Note that if the reheating temperature after inflation is low, those mirror particles might not be produced abundantly, since these heavy mirror neutrinos can be in thermal equilibrium only above the temperature, ∼ (v R /v L · 10 −6 ) 4/3 × 100 TeV. In such a case, the above bound could be much relaxed.

Summary
We have proposed an extended Standard Model (SM), mainly motivated by the strong CP problem and dark matter. The model has an extended gauge symmetry at the high scale, SU (3) c × SU (2) L × U (1) L × SU (2) R × U (1) R , which breaks into the SM at the low scale. To achieve the symmetry, mirror quarks, leptons, gauge bosons and Higgs are added. A dark matter particle serves as another portal to connect two sectors, which can also help to avoid the abundance problem of the lightest stable mirror quarks and leptons. We has investigated various phenomenologies in this model, including direct detection of dark matter, flavor physics, collider physics and cosmological effects. One special prediction of this model is the ratio of mirror particles' masses, which might be probed at future collider searches. At this moment, the LHC data has already constrained the mass of the lightest mirror quark, O(600)GeV. This indicates that the breaking scale of the mirror section should be larger than 7 × 10 7 GeV. In addition, we have numerically shown that if the observed DM abundance is saturated with the WIMP DM in our model, mirror up quark has to be lighter than about 3 TeV in order to avoid various bounds. The DM mass is also around the mirror up quark mass or the mirror electron mass. Even if we take into account the other effect such as the contribution of the Higgs portal coupling, our conclusion is not changed as discussed in Appendix B. Then, we conclude that the parity symmetry breaking scale should be 4 × 10 8 GeV corresponding to the upper bound on the mirror up quark mass. In the parameter region, the mirror electron resides around 200 GeV-400 GeV. Such mass region requires more detailed analysis taking into account the constraints from the flavor physics, collider physics and the EW precision measurement. Another observable is the change of effective relativistic degree of freedom by right-handed neutrinos, which is in the reach of future CMB experiments.
For Higgs portal process, annihilation is s-wave dominant, while for mirror fermion exchanging it is p-wave dominant. Since there is no interference between different partial waves, the annihilation cross section can be separated into two contributions #6 , σv a h λ 2 where λ hX and λ are a Higgs portal coupling and a X-ψ-ψ Yukawa coupling given in Eq. (49), respectively. This is valid except for coannihilation region. Suppose that the Higgs portal contribution is r times larger than that of mirror fermion exchanging at freeze-out, i.e., a h λ 2 hX = rb f v 2 |λ| 4 , the cross section is rewritten as To explain the DM abundance, λ should satisfy the equation, We define λ satisfying Eq.(109) as λ(r). It is easy to see that λ(r) is related to λ(0) as This means for a nonzero Higgs portal coupling, λ required to explain the DM abundance is smaller by a factor, 1/(1 + r) 1/4 , than the vanishing case. The SI cross section of WIMP DM and nuclei scattering is given by where µ = m A m X /(m A + m X ) denotes reduced mass of DM and nucleus and we assume the symmetric relic, i.e. Ω DM = Ω DM . f N,e contains only C S and C T and f N,o does only C V in Eq. (53). Note that the λ coupling mainly generates f N,o through photon and Z boson exchanging, while the Higgs portal interaction only f N,e . Then, in our model, it takes a form of σ SI (r) = C HP λ 2 hX + C F P |λ(r)| 4 .
From this equation, if σ SI (r) < σ SI (0) = C F P |λ(0)| 4 , we conclude the Higgs portal interaction can relax the direct detection constraints. Using Eqs.(107), (109) and (110), we obtain σ SI (r) = r 1 + r σ HP + 1 1 + r σ SI (0), #6 For m X 1 TeV, annihilation via Higgs portal is dominated by XX † → W W, ZZ, hh processes, so that there is no large interference in XX † → tt process. Hence, in such a mass region, once we replace like b f v 2 → (a f + b f v 2 ), discussion below can be applied straightforwardly even in top-philic case.
where σ HP C HP × 10 −9 [GeV −2 ] a h (114) corresponds to the SI cross section for the pure Higgs portal DM scenario and is estimated as σ HP 1.7 × 10 −9 [pb] when m X O(TeV). Thus, the SI cross section is reduced only if a ≡ σ HP σ SI (0) < 1.
We would like to point out that the modified cross section is bounded: This indicates that a mass region already excluded in the Higgs portal scenario is not rescued even if we introduce Higgs portal interaction. The current XENON1T result rules out the DM mass 2 TeV for complex scalar DM in the Higgs portal scenario. For example, in charm-philic case, σ SI (0) is for m X = 3 TeV and m u = 4 TeV. This is comparable with the Higgs portal one, and hence direct detection bound is not relaxed in this parameter region. The value of λ is reduced to, e.g.
This will still suffer from Landau pole constraint, however. Thus, we do not expect improvement in charm-philic case.
Therefore, in the top-philic case, we can expect some reduction of the SI cross section by introducing a nonzero λ hX , but it is not enough to evade various constraints.