Chiral Composite Asymmetric Dark Matter

In this subsection, we review the composite ADM model in Ref. Ibe et al. (2018). The model is based on $SU(3)_{D}\times U(1)_{D}$ gauge symmetry with $N_{f}=2$. The dark quarks are in the fundamental representations of $SU(3)_{D}$ and have $U(1)_{D}$ and $U(1)_{B-L}$ charges like the up-type and the down-type quarks in the visible sector. In Tab. 1, we show the charge assignment of the dark quarks, where all the (anti-)quarks are given as the left-handed Weyl fermions. Hereafter, we put the primes on the fields and the parameters in the dark sector. As is evident from the table, $SU(3)_{D}$ and $U(1)_{D}$ symmetries are analogous to the visible QCD and QED, and hence, we call them dark QCD and dark QED, respectively. Incidentally, let us emphasize that the absence of the dark weak interaction and the dark neutrinos is a quite important feature for successful cosmology because the dark neutrinos, if present, can be the lightest dark sector particle and lead to too large $N_{\mathrm{eff}}$.

Since the dark quarks are in the vector-like representation, they have masses,

$\displaystyle\mathcal{L}_{\mathrm{mass}}=m_{U^{\prime}}\bar{U}^{\prime}U^{\prime}+m_{D^{\prime}}\bar{D}^{\prime}D^{\prime}+\mathrm{h.c.}$

(1)

Here, we assume that the dark quark mass parameters $m_{U^{\prime},D^{\prime}}$ are smaller than the dynamical scale of dark QCD, $\Lambda^{\prime}_{D}$. Below $\Lambda^{\prime}_{D}$, the dark quarks are confined into the dark mesons and dark baryons as in QCD. The lightest dark mesons are given by

$\displaystyle\pi^{\prime 0}\propto U^{\prime}\bar{U}^{\prime}-D^{\prime}\bar{D}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \pi^{\prime+}\propto U^{\prime}\bar{D}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \pi^{\prime-}\propto D^{\prime}\bar{U}^{\prime}\ ,$

(2)

which are analogous to the pions in QCD. The lightest dark baryons are also given by,

$\displaystyle p^{\prime}\propto U^{\prime}U^{\prime}D^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{p}^{\prime}\propto\bar{U}^{\prime}\bar{U}^{\prime}\bar{D}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ n^{\prime}\propto U^{\prime}D^{\prime}D^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{n}^{\prime}\propto\bar{U}^{\prime}\bar{D}^{\prime}\bar{D}^{\prime},$

(3)

which are analogues of the proton and the neutron, respectively. By the assumption of $m_{U^{\prime},D^{\prime}}\ll\Lambda^{\prime}_{D}$, the dark pion are much lighter than the dark baryons as in the case of QCD.

Since the $B-L$ symmetry is well approximate symmetry within the dark sector, the lifetime of the dark baryons can be much longer than the age of the Universe Fukuda et al. (2015). The dark baryons annihilate into the dark pions, and hence, the relic density of the symmetric component is subdominant due to the large annihilation cross-section. Hence, the present dark baryon density is dominated by the asymmetric component as in the case of the visible baryon. For a given ratio of the $B-L$ asymmetry in the visible sector to that of the dark sector, $A_{\mathrm{DM}}/A_{\mathrm{SM}}$, the observed dark matter abundance is explained for

$\displaystyle m_{\mathrm{DM}}\simeq\frac{\Omega_{\mathrm{DM}}}{\Omega_{\mathrm{B}}}\frac{A_{\mathrm{B}}}{A_{\mathrm{SM}}}\frac{A_{\mathrm{SM}}}{A_{\mathrm{DM}}}\times m_{\mathrm{N}}\ ,$

(4)

with the observed values, $\Omega_{\mathrm{DM}}h^{2}=0.120\pm 0.001$ and $\Omega_{B}h^{2}=0.0224\pm 0.0001$ Aghanim et al. (2020). Here, $A_{\mathrm{B}}/A_{\mathrm{SM}}=30/97$ is the ratio between the baryon asymmetry and the $B-L$ asymmetry in visible sector Harvey and Turner (1990),²²2We assume that the top quark decouples earlier than the sphaleron processes Ibe et al. (2012). and $m_{\mathrm{N}}\simeq 0.94\,$GeV is the nucleon mass. Thus, for $A_{\mathrm{DM}}/A_{\mathrm{SM}}=\order{1}$  the dark matter mass should be of $\order{1}$ GeV. Such baryon masses are achieved when the dynamical scale of the dark QCD is of $\order{1}$ GeV.

For example, in a class of models in which the $B-L$ symmetry in the visible and the dark sectors are thermally distributed through the higher dimensional operator, the ratio is given by $A_{\mathrm{DM}}/A_{\mathrm{SM}}=22N_{f}/237$ Fukuda et al. (2015) (see also the Appendix B).³³3Here, we neglected the conservation of the $U(1)_{D}$ charge. The correct ratio is $66/395$ for the charge assignment in Tab. 1, which is not very different from $22N_{f}/237$ with $N_{f}=2$. In this case, the mass of the DM (i.e. the dark baryons) is required to be

$\displaystyle m_{\mathrm{DM}}\simeq\frac{17}{N_{f}}\,\mathrm{GeV}\ ,$

(5)

where $N_{f}=2$ is the minimal choice.

The dark pions are also produced abundantly in the thermal bath, which could cause cosmological problems if they remain in the present Universe. Those problems are solved by the presence of the dark photon lighter than a half of the pion masses. The dark neutral pion immediately decays into a pair of the dark photons due to the chiral anomaly. The stable dark charged pions also annihilate into a pair of the dark photons, and hence, their relic abundance is subdominant.

The dark photon eventually decays into a pair of the electron and the positron through the kinetic mixing with the visible photon given by,

$\displaystyle\mathcal{L}_{\gamma^{\prime}}=-\frac{1}{4}F^{\prime\mu\nu}F^{\prime}_{\mu\nu}+\frac{\epsilon}{2}F_{\mu\nu}F^{\prime\mu\nu}+\frac{1}{2}m^{2}_{\gamma^{\prime}}A^{\prime}_{\mu}A^{\prime\mu}\ .$

(6)

Here, $m_{\gamma^{\prime}}$ denotes the mass of the dark photon, $A_{\mu}^{\prime}$, $\epsilon$ is the kinetic mixing parameter, and $F_{\mu\nu}$ and $F_{\mu\nu}^{\prime}$ are the field strengths of the visible and the dark photons, respectively. As discussed in Ref. Ibe et al. (2018), the dark photon successfully transfers the excessive energy/entropy in the dark sector into the visible sector for $m_{\gamma^{\prime}}=\order{10\mbox{--}100}$ MeV and $\epsilon=\order{10^{-8}\mbox{--}10^{-11}}$.⁴⁴4See Refs. Ibe et al. (2019a, b) for a possible origin of the tiny kinetic mixing parameters.

In the above discussion, we have not specified the origin of the dark photon mass. The simplest possibilities are to introduce a Higgs boson or to assume the Stückelberg model Ruegg and Ruiz-Altaba (2004). In these models, however, we require parameter tuning so that the dark photon mass is of $\order{10\mbox{--}100}$ MeV while the DM mass is of $\order{1}$ GeV. To avoid such parameter tuning, we apply a mechanism which generates a dark photon mass due to strong dynamics Harigaya and Nomura (2016); Co et al. (2017). In this subsection, we review the dynamical generation of the dark photon mass.

Let us continue to consider a model with $N_{f}=2$ which has $SU(3)_{D}\times U(1)_{D}$ gauge symmetry as in the previous subsection. The charge assignment of $U(1)_{D}$ is, on the other hand, changed to the one in Tab. 2. For $0<a<1$, the $U(1)_{D}$ gauge symmetry is no more vector-like symmetry, and hence, the mass terms in Eq. (1) are now forbidden. The $U(1)_{D}$ gauge symmetry is free from gauge anomalies. The global $U(1)_{B-L}$ symmetry, on the other hand, has the global anomaly of $U(1)_{B-L}\times U(1)_{D}^{2}$, although this does not affect the ADM scenario unless there is a dark helical magnetic field in the Universe (see, e.g., Ref. Kamada and Long (2016)).

The assumption of the chiral $U(1)_{D}$ is crucial for the dynamical breaking of $U(1)_{D}$, since the vector-like symmetry cannot be broken spontaneously by strong dynamics Vafa and Witten (1984). Note also that the $U(1)_{D}$ gauge symmetry explicitly breaks the $SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$ flavor symmetry of the dark quarks into the third component of the dark isospin symmetry, $U(1)_{3}^{\prime}$. Hereafter, $I^{\prime}_{3}$ refers to the charge under $U(1)_{3}^{\prime}$. The $SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$ flavor symmetry remains an approximate symmetry as long as the $U(1)_{D}$ gauge interaction is perturbative.

Below $\Lambda_{D}^{\prime}$, the dark quark bilinears condense as follows,

$\displaystyle\Braket{U^{\prime}\bar{U}^{\prime}+U^{\prime\dagger}\bar{U}^{\prime\dagger}}=\Braket{D^{\prime}\bar{D}^{\prime}+D^{\prime\dagger}\bar{D}^{\prime\dagger}}=\order{\Lambda_{D}^{\prime 3}}\ .$

(7)

The condensate in this channel is expected to be favored than other channels such as $\langle U^{\prime}\bar{D}^{\prime}\rangle$ since this channel has the smallest $U(1)_{D}$ charge, that is $|1-a|<|1+a|$, for $0<a<1$ Harigaya and Nomura (2016). These condensations spontaneously break the $U(1)_{D}$ gauge symmetry. Besides, they also break the approximate $SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$ flavor symmetry into the diagonal subgroup, $SU(2)^{\prime}_{V}$. On the other hand, $U(1)_{B-L}$ is not broken by the condensations, and hence, $U(1)_{B-L}$ and $U(1)_{3}^{\prime}$ are exact (accidental) symmetries up to $U(1)_{D}$ anomaly.⁵⁵5The appearance of the accidental symmetries is generic advantage of the composite dark matter models Antipin et al. (2015); Bottaro et al. (2021).

Associated with spontaneous breaking of $SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$ into $SU(2)^{\prime}_{V}$, there are three pseudo Nambu-Goldstone (NG) bosons. The low energy effective theory of the NG bosons is well described by the matrix-valued $SU(2)$ field,

$\displaystyle U(x)=\mathrm{exp}\Biggl{[}\frac{i}{f_{\pi}^{\prime}}\sum_{i=1}^{3}\pi^{\prime}_{i}(x)\sigma_{i}\Biggr{]}\ ,$

(8)

where $f_{\pi}^{\prime}$ denotes the dark pion decay constant, $\pi^{\prime}_{i}(x)\,(i=1,2,3)$ are three Nambu-Goldstone (NG) bosons and $\sigma_{i}\,(i=1,2,3)$ are the Pauli matrices.⁶⁶6We take the normalization of Eqs. (8) and (9) so that the corresponding $f_{\pi}$ in the QCD is $f_{\pi}\simeq 93$ MeV. Of these three NG bosons, $\pi^{\prime}_{3}$ becomes the longitudinal component of the dark photon, which is evident that $U(1)_{D}$ is realized by the shift of $\pi^{\prime}_{3}$ at around $\vec{\pi^{\prime}}=0$. Hereafter, we call the two remaining NG bosons, $\pi^{\prime}\equiv(\pi^{\prime}_{1}+i\pi^{\prime}_{2})/\sqrt{2}$ (and its complex conjugate ${\pi}^{\prime\dagger}$), the dark pions.

The kinetic term and the $U(1)_{D}$ gauge interaction of the dark pion is described by

$\displaystyle\mathcal{L}=\frac{f^{\prime 2}_{\pi}}{4}\mathrm{tr}[(D_{\mu}U)(D^{\mu}U)^{\dagger}]\ ,$

(9)

where the covariant derivative of $U(x)$ is given by,

$\displaystyle D_{\mu}U=\partial_{\mu}U(x)-ie_{D}A_{\mu}^{\prime}\sigma_{3}U(x)+iae_{D}A_{\mu}^{\prime}U(x)\sigma_{3}\ .$

(10)

Here, $e_{D}$ is the gauge coupling constant of $U(1)_{D}$. In the “unitary gauge”, $\pi^{\prime}_{3}=0$, we obtain interactions between the dark pion and the dark photon,

$\displaystyle\mathcal{L}=(D_{\mu}\pi^{\prime})^{\dagger}(D^{\mu}\pi^{\prime})-e^{2}_{D}(1-a)^{2}A_{\mu}^{\prime}A^{\prime\mu}\pi^{\prime\dagger}\pi^{\prime}+\frac{1}{2}e_{D}^{2}(1-a)^{2}f_{\pi}^{\prime 2}A_{\mu}^{\prime 2}+\cdots\ ,$

(11)

where the ellipses denote the higher dimensional terms suppressed by $f_{\pi^{\prime}}$. We introduced the “covariant” derivative of $\phi$,

$\displaystyle D_{\mu}\pi^{\prime}=\partial_{\mu}\pi^{\prime}+ie_{D}(1+a)A_{\mu}^{\prime}\pi^{\prime}\ .$

(12)

The $U(1)_{D}$ invariance of Eq. (11) is not manifest due to the non-linear realization of $U(1)_{D}$, although the effective theory in Eq. (9) is manifestly $U(1)_{D}$ invariant.

The third term of Eq. (11) gives the dark photon mass,

$\displaystyle m_{\gamma^{\prime}}=e_{D}(1-a)f_{\pi}^{\prime}\simeq\frac{\sqrt{3}}{4\pi}e_{D}(1-a)m_{\rho^{\prime}}\ ,$

(13)

In the final expression, we have used the naive dimensional analysis between the (dark) pion decay constant and the (dark) rho meson mass Manohar and Georgi (1984),

$\displaystyle f_{\pi}^{\prime}\simeq\frac{\sqrt{N_{c}}}{4\pi}m_{\rho^{\prime}}\ ,$

(14)

with $N_{c}=3$. Here, $m_{\rho^{\prime}}$ is the mass of the dark rho meson. The dark photon becomes massless for $a=1$, which corresponds to the vector-like $U(1)_{D}$.

As the $U(1)_{D}$ gauge interaction breaks the $SU(2)^{\prime}_{L}\times SU(2)^{\prime}_{R}$ symmetry explicitly, the corresponding NG boson $\pi^{\prime}$ obtains the non-vanishing mass. Following Das et al. (1967) (see also Cheng and Li (1984)), we obtain the dark pion mass squared,⁷⁷7This expression is four times larger than that in Ref. Harigaya and Nomura (2016).

$\displaystyle m_{\pi^{\prime}}^{2}\simeq\frac{3a\log{2}}{2\pi^{2}}e_{D}^{2}m_{\rho^{\prime}}^{2}\,.$

(15)

Here, we neglected the dark photon mass whose effects are suppressed by $\order{m_{\gamma^{\prime}}^{2}/m_{\rho^{\prime}}^{2}}$. The dark pion becomes massless for $a=0$, where the $U(1)_{D}$ gauge symmetry does not break $SU(2)^{\prime}_{R}$ explicitly. By comparing Eqs. (13) and (15), we find that the condition $a\gtrsim 0.2$ is required for the dark pion mass to be larger than the dark photon mass.

Some constraints on the dark photon/pion masses put bounds on $a$ and $e_{D}$. In Fig. 1, we show the viable parameter region. The figure shows that the requirement $m_{\pi^{\prime}}>m_{\gamma^{\prime}}$ is achieved for $a\gtrsim 0.13$ (outside the green shaded region). We also show the lower limits on $m_{\gamma^{\prime}}$ from the effective number of neutrino degrees of freedom, $N_{\mathrm{eff}}$ Ibe et al. (2020a) as blue/orange shaded regions. The blue shaded region corresponding to $m_{\mathrm{\gamma^{\prime}}}\lesssim 8.5$ MeV is excluded by the $N_{\mathrm{eff}}$ constraint from the Planck observation of the cosmic microwave background (CMB) Aghanim et al. (2020) for $\epsilon\gtrsim 10^{-9}$. The orange one corresponding to $m_{\mathrm{\gamma^{\prime}}}\lesssim 17$ MeV shows the future sensitivity of the stage-IV CMB experiment Abazajian et al. (2016).

By combining the ideas in Sec. A and Sec. B, we construct the chiral composite ADM model in which the dark photon mass is generated dynamically. In the model with $N_{f}=2$ so far considered, the lightest dark baryons are

$\displaystyle p^{\prime}\propto U^{\prime}U^{\prime}D^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{p}^{\prime}\propto\bar{U}^{\prime}\bar{U}^{\prime}\bar{D}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ n^{\prime}\propto U^{\prime}D^{\prime}D^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{n}^{\prime}\propto\bar{U}^{\prime}\bar{D}^{\prime}\bar{D}^{\prime}\leavevmode\nobreak\ ,$

(16)

as in QCD. Due to the non-trivial baryon charges, they are stable and eventually become DM. The mass partner of $p^{\prime}$ is $\bar{p}^{\prime}$, and that of $n^{\prime}$ is $\bar{n}^{\prime}$. This combination results from the $U(1)^{\prime}_{3}$ and the $U(1)_{B-L}$ symmetries. Besides, the masses of the dark proton and the dark neutron are identical due to the charge conjugation symmetry which interchanges $U^{\prime}$ and $D^{\prime}$. Therefore, both the dark proton and the dark neutron become dark matter.

In summary, we obtain the chiral composite ADM model based on the charge assignment in Tab. 2, which naturally provides the desirable spectrum,

$\displaystyle m_{\gamma^{\prime}}<m_{\pi^{\prime}}<m_{\mathrm{DM}}\ .$

(17)

In this model,

• •

  The dark baryons are stable due to $B-L$ symmetry.

• •

  The dark baryon density is dominated by the asymmetric components due to the large annihilation cross section of the dark baryons into the dark pions.

• •

  The chiral $U(1)_{D}$ gauge symmetry is broken dynamically, whose mass is suppressed compared with the dark baryons.

• •

  The dark pion mass is generated by the radiative correction of the $U(1)_{D}$ interaction. The mass can be larger than the dark photon mass, and hence, the dark pion annihilates into the dark photon pairs efficiently.⁸⁸8In the present model, there is no “dark neutral pion” as it is absorbed by the dark photon. Thus, there is no need to assume that the dark pion mass is twice larger than the dark photon mass.

For $A_{\mathrm{DM}}/A_{\mathrm{SM}}=\order{1}$, the dark matter density is reproduced by $m_{\mathrm{DM}}=\order{1}$ GeV, with which the masses of the dark photon and the dark pion are predicted in the sub-GeV due to the dynamical symmetry breaking.

For successful ADM models, we need a portal interaction with which the $B-L$ asymmetry is thermally distributed between the visible and the dark sectors. One of such portal interactions can be given by higher-dimensional operators Ibe et al. (2012); Fukuda et al. (2015),

$\displaystyle\mathcal{L}_{B-L\,\mathrm{portal}}=\frac{1}{M_{*}^{n}}\mathcal{O}_{D}LH\ +\mathrm{h.c.}$

(18)

Here, $L$ and $H$ denote the lepton and the Higgs doublets in the visible sector. $\mathcal{O}_{D}$ is a $B-L$ charged but $U(1)_{D}$ neutral operator in the dark sector with the mass dimension $d_{D}=n+3/2$. $M_{*}$ encapsulates the energy scale of the portal interactions. With this portal interaction, the visible and the dark sectors are in the thermal equilibrium at the high temperature. The $B-L$ asymmetry is also thermally distributed through the same operators. The portal interaction eventually decouples at the temperature,

$\displaystyle T_{D}\sim M_{*}\left(\frac{M_{*}}{M_{\mathrm{Pl}}}\right)^{1/(2n-1)}\ ,$

(19)

where $M_{\mathrm{Pl}}\simeq 2.4\times 10^{18}$ GeV is the reduced Planck mass scale. Below the decoupling temperature, the $B-L$ asymmetries in the two sector are conserved separately.

In the model in the previous subsection, however, there is no good candidate for $\mathcal{O}_{D}$ which is $B-L$ charged but neutral under the $SU(3)_{D}\times U(1)_{D}$ gauge symmetry. To allow the $B-L$ portal interaction in Eq. (18), we extend the composite ADM model with an additional generation of dark quarks so that $N_{f}=3$.⁹⁹9We may instead extend the non-Abelian gauge group to $SU(2)_{D}$ while keeping $N_{f}=2$. The charge assignment of the dark quarks under the gauge and the global symmetries are given in Tab. 3.

In this model, the third flavor, $(S^{\prime},\bar{S}^{\prime})$, is vector-like, and hence, it has a mass $m_{S^{\prime}}$, where we assume $m_{S^{\prime}}<\Lambda_{D}^{\prime}$. In the presence of the third flavor dark quark, the $B-L$ portal interaction can be given by,

$\displaystyle\mathcal{L}_{B-L\,\,\mathrm{portal}}=\frac{1}{M_{*}^{3}}({U}^{\prime}{D}^{\prime}{S}^{\prime})LH+\frac{1}{M_{*}^{3}}(\bar{U}^{\prime\dagger}\bar{D}^{\prime\dagger}S^{\prime})LH+\mathrm{h.c.}$

(20)

For the $B-L$ asymmetry generation mechanism, we assume, for example, the thermal leptogenesis Fukugita and Yanagida (1986) as in Ref. Ibe et al. (2018). In the thermal leptogenesis, the reheating temperature, $T_{R}$, after inflation is required to be higher than, $T_{R}\gtrsim 10^{9.5}$ GeV Giudice et al. (2004); Buchmuller et al. (2005). Thus, for the scenario to be successful, we require $T_{D}\lesssim T_{R}$, which reads,

$\displaystyle M_{*}\lesssim T_{R}\left(\frac{M_{\mathrm{Pl}}}{T_{R}}\right)^{1/2n}\lesssim 10^{11}\,\mathrm{GeV}\times\left(\frac{T_{R}}{10^{9.5}\,\mathrm{GeV}}\right)^{5/6}\ ,$

(21)

where we have used $n=3$ in the last inequality. In this case, we assume that the portal interaction is generated by a new sector at $M_{*}\simeq 10^{11}$ GeV (see Ref. Ibe et al. (2018) for details).

Our portal mechanism works with other types of baryogenesis in the visible sector. Again, $T_{D}$ should be lower than the temperature of the completion of the baryogenesis. In addition, when the baryogenesis does not generate the lepton asymmetry directly, $T_{D}$ is required to be lower than the temperature $T_{\mathrm{sph}}\sim 10^{12}$ GeV, below which the sphaleron process is in equilibrium (see e.g. Ref. Moore (2000)). Other leptogenesis in the visible sector which completes above $T_{D}$ also works. However, the numerical value of the dark sector asymmetry will be altered from the one used in the present paper if $T_{\mathrm{sph}}<T_{D}$.

For $m_{S}^{\prime}\lesssim\Lambda_{D}^{\prime}$, the dark quark bilinears condense as,

$\displaystyle\Braket{U^{\prime}\bar{U}^{\prime}+U^{\prime\dagger}\bar{U}^{\prime\dagger}}=\Braket{D^{\prime}\bar{D}^{\prime}+D^{\prime\dagger}\bar{D}^{\prime\dagger}}=\Braket{S^{\prime}\bar{S}^{\prime}+S^{\prime\dagger}\bar{S}^{\prime\dagger}}=\order{\Lambda_{D}^{\prime 3}}\ .$

(22)

The condensates spontaneously break the $U(1)_{D}$ gauge symmetry and the approximate $SU(3)^{\prime}_{L}\times SU(3)^{\prime}_{R}$ symmetry to the diagonal subgroup $SU(3)^{\prime}_{V}$. Note that the two components of the Cartan subgroup of $SU(3)^{\prime}_{V}$, $U(1)_{3}^{\prime}$ and $U(1)_{8}^{\prime}$, are exact (accidental) symmetry up to $U(1)_{D}$ anomaly and remain unbroken by the condensates in Eq. (22). As in the model with $N_{f}=2$, the NG boson corresponds to $\pi^{\prime}_{3}$ becomes the longitudinal component of the $U(1)_{D}$ gauge boson.

As in QCD, the dark quarks are confined into hadrons, and the lightest baryons and the NG bosons form the octet representations of $SU(3)^{\prime}_{V}$,

$\displaystyle B_{\alpha}^{\prime}\!=\!\left(\begin{array}[]{ccc}\Sigma^{\prime 3}_{\alpha}+\Lambda^{\prime}_{\alpha}/\sqrt{3}&\sqrt{2}\Sigma^{\prime 1}_{\alpha}&\sqrt{2}p_{\alpha}^{\prime}\\ \sqrt{2}\Sigma^{\prime 2}_{\alpha}&-\Sigma^{\prime 3}_{\alpha}+\Lambda^{\prime}_{\alpha}/\sqrt{3}&\sqrt{2}n_{\alpha}^{\prime}\\ \sqrt{2}\Xi^{\prime 2}_{\alpha}&\sqrt{2}\Xi^{\prime 1}_{\alpha}&-2\Lambda^{\prime}_{\alpha}/\sqrt{3}\end{array}\right),\,\,\,M^{\prime}\!=\!\left(\begin{array}[]{ccc}\eta^{\prime}/\sqrt{3}&\sqrt{2}\pi^{\prime}&\sqrt{2}K^{\prime 1}\\ \sqrt{2}\pi^{\prime\dagger}&\eta^{\prime}/\sqrt{3}&\sqrt{2}K^{\prime 2}\\ \sqrt{2}K^{\prime 1\dagger}&\sqrt{2}K^{\prime 2\dagger}&-2\eta^{\prime}/\sqrt{3}\end{array}\right)\ ,$

(29)

where we have taken the unitary gauge, i.e., $\pi^{\prime}_{3}=0$. The index $\alpha$ denotes the Weyl spinor components. The names of the dark baryons and the dark mesons are after the corresponding baryons and the mesons in the visible sector.¹⁰¹⁰10The dark $\eta^{\prime}$ corresponds to $\eta$ in the visible sector. The $U(1)_{D}$ charges of them are not parallel with the $U(1)_{\mathrm{QED}}$ charges in the visible sector. The dark baryons are also associated with their antiparticles, $\bar{B}_{\alpha}$. The mesons $\pi^{\prime}$ and $K^{\prime 1,2}$ are complex scalars, while $\eta^{\prime}$ is a pseudo scalar.

As in the model with $N_{f}=2$, all the physical NG bosons become massive. The $\pi^{\prime}$ mass is from the $U(1)_{D}$ interaction as in Eq. (15). The NG bosons which have the $S$ components obtain masses of $\order{\sqrt{m_{S^{\prime}}\Lambda_{D}^{\prime}}}$ due to the explicit mass term of $(S^{\prime},\bar{S}^{\prime})$.¹¹¹¹11Due to the charge conjugation symmetry which exchanges $U^{\prime}$ and $D^{\prime}$, the masses of $K^{\prime 1,2}$ are identical. In the following, we assume that they are slightly heavier than the dark pion, so that they annihilate into the dark pions very efficiently. Note that all the NG bosons other than $\eta^{\prime}$ are stable when no NG bosons are twice heavier than the other NG bosons.¹²¹²12When $\eta^{\prime}$ is twice heavier than $\gamma^{\prime}$, it decays into a pair of the dark photons. For a lighter $\eta^{\prime}$, it decays into $\gamma^{\prime}+e^{+}+e^{-}$ through the kinetic mixing.

The leading mass term of the baryons is from

	$\displaystyle\mathcal{L}_{\mathrm{mass}}$	$\displaystyle\simeq\frac{1}{2}m_{B}\mathop{\rm tr}[BU\bar{B}U^{\dagger}]+\mathrm{h.c.}\ ,$		(30)
		$\displaystyle\simeq m_{B}(p^{\prime}\bar{p}^{\prime}+n^{\prime}\bar{n}^{\prime}+\Lambda^{\prime}\bar{\Lambda}^{\prime}+\sum_{i=1}^{3}\Sigma^{\prime i}\bar{\Sigma}^{\prime i}+\sum_{i=1}^{2}\Xi^{\prime i}\bar{\Xi}^{\prime i})+\mathrm{h.c.}\ ,$		(31)

where

$\displaystyle U(x)=\exp\left[\frac{i}{f_{\pi}^{\prime}}M(x)\right]\ .$

(32)

The term in Eq. (30) is invariant under $U(1)_{D}\times SU(3)^{\prime}_{L}\times SU(3)^{\prime}_{R}$. Note that the dark baryon masses are not identical due to the $U(1)_{D}$ gauge interaction and the mass of $(S^{\prime},\bar{S}^{\prime})$.