Communication with SIMP dark mesons via Z'-portal

We consider a consistent extension of the SIMP models with dark mesons by including a dark U(1)_D gauge symmetry. Dark matter density is determined by a thermal freeze-out of the $3\to2$ self-annihilation process, thanks to the Wess-Zumino-Witten term. In the presence of a gauge kinetic mixing between the dark photon and the SM hypercharge gauge boson, dark mesons can undergo a sufficient scattering off the Standard Model particles and keep in kinetic equilibrium until freeze-out in this SIMP scenario. Taking the SU(N_f)xSU(N_f)/SU(N_f) flavor symmetry under the SU(N_c) confining group, we show how much complementary the SIMP constraints on the parameters of the dark photon are for current experimental searches for dark photon.


I. INTRODUCTION
Various evidences of dark matter (DM) imply that fundamental particles and interactions need to be extended beyond the Standard Model (SM) [1]. One of the appealing suggestions is the thermal DM scenario, where the DM relic density is determined through the freeze-out of the DM number changing process. Weakly Interacting Massive Particle (WIMP) provides the most popular thermal DM scenario, in which the annihilation of a DM pair into a pair of SM particles is responsible for the freeze-out. Since WIMP mass is of order weak scale for the effective coupling of α eff ∼ O(10 −2 ), 'WIMP miracle' has been the mainstream for thermal DM studies, corroborating the expectation of finding new physics at the weak scale in the solutions for gauge hierarchy problem.
Another interesting proposal for thermal (pseudo-) scalar DM has been recently made under the name of Strongly Interacting Massive Particle (SIMP) [2], explaining DM relic density through the freeze-out of 3 → 2 self-annihilation. The DM self-interaction is motivated by potential small-scale problems [3], although it is strongly constrained by bullet cluster [4] and simulations on halo shape [5]. As a result, the SIMP scenario predicts dark matter with dimensionless self-interacting coupling of order one and mass in the 0.1 − 1 GeV range, which has not been explored seriously so far.
The SIMP scenario requires the interaction between dark and SM sectors in the form of scattering for dark matter to be in kinetic equilibrium with the SM particles, without altering the structure formation [6]. Since such an inter-sector interaction also leads to the DM annihilation into SM particles, the inter-sector interaction strength is bounded from above for the dominance of 3 → 2 self-annihilation, if combined, resulting in n DM σv ann < n 2 DM σv 3→2 < n SM σv scatt , at the freeze-out temperature. Taking other constraints from ground-based experiments into account in addition, we can make quite a concrete prediction on the parameters of a specific SIMP model.
In this article, we consider a SIMP model with dark mesons suggested in Ref. [7], where the 5-point interactions between dark mesons for 3 → 2 annihilation come from the leading interactions of the Wess-Zumino-Witten (WZW) term [8,9]. From the model point of view, the WZW term is interesting because it encodes various aspects on dark sector, namely, the WZW term exists only for a specific flavor symmetry of light dark quarks, depending on its spontaneous breaking pattern [10], and it contains color number as a topological index [9]. As an inter-sector interaction, we consider the hidden valley scenario [11], that some heavy dark sector particle has a renormalizable coupling to the mediator particle that communicates between DM and SM particles. Higgs-portal interaction would be a natural candidate, but it is not enough for a sufficiently large DM-SM particle scattering at freezeout temperature due to small Yukawa couplings of the light SM fermions. Thus, we study the case with a gauge kinetic mixing [12], that is the renormalizable and gauge invariant interaction between SM hypercharge U(1) Y and dark sector U(1) D with a dimensionless coupling. When the dark U(1) gauge coupling is not too tiny, dark meson annihilation into dark photons could easily dominate the annihilation process of dark matter. But, we can forbid it by taking the dark photon to be heavier than dark mesons. In this case, the gauge kinetic mixing plays a role of 'hidden valley' in the SIMP scenario.
In Sec. II, we briefly review dark mesons in the SIMP scenario, the abundance of which is frozen out by the WZW term. In Sec. III, we discuss properties of the dark U(1) gauge symmetry, U(1) D , that is compatible with both the WZW term and the SIMP scenario.
By taking SU(N c ) as an example of confining gauge group, in Sec. IV, we present a viable parameter range for the dark photon mass and the strength of the gauge kinetic mixing.
Finally, conclusions are drawn.

II. SIMP DARK MESONS AND THE WZW TERM
Dark meson appears as a composite state of dark quarks in models with dark confining gauge group G c and it has several interesting properties [13]. The lightest mesons are interpreted as pseudo-Goldstone bosons from a spontaneous breaking of (accidental) flavor symmetry, which guarantees their stability. Whereas flavor symmetry is broken by higher dimensional operators, due to the compositeness of dark mesons, their decays induced by higher dimensional operator is suppressed, as compared to those of a fundamental scalar, so the stability of dark mesons is easier to achieve. For instance, in the presence of parity conservation in the inter-sector interaction, dark mesons can be unstable by the decay into two photons, due to the following higher dimensional operator, where M is the scale at which an explicit breaking of flavor symmetry occurs, T a is the generator of flavor symmetry, Λ is the confining scale, and F is the dark meson decay constant of O(Λ). Note that this operator is dimension-7, rather than dimension-5, which is the dimension of the corresponding operator for a fundamental pseudo-scalar DM. Then, the lifetime of dark mesons is given by Γ −1 = πF 2 M 6 /(m 3 Λ 6 ), and, for GeV-scale dark mesons and a similar confining scale, it is longer than the age of universe, 6.6 × 10 41 GeV −1 , as long as M is larger than 10 7 GeV. Moreover, the interactions between dark mesons induce the DM self-scattering, which provides a solution to the small-scale problems such as 'core-cusp' or 'too-big-to-fail' problems.
In the SIMP scenario proposed in Ref. [2], the DM relic density can be explained from the freeze-out of dark mesons in the presence of their 5-point self-interactions, provided by the WZW term [7], The WZW term is the outcome of a specific flavor symmetry G f and its spontaneous breaking to the subgroup H for a given confining gauge group, relying on a nontrivial fifth homotopy group, π 5 (G f /H) = Z [10]. The manifest non-chiral global symmetry H is unbroken if it is respected by dark quark masses [14]. As a consequence, degenerate dark quark masses m q , or degenerate dark meson mass m 2 π = 8(Λ 3 /F 2 )m q guarantee the existence of the WZW term.
Here, we quote the results obtained in Ref. [7], which will be useful for the discussion hereafter. The 3 → 2 annihilation cross section is calculated from the WZW term to be where T F is the freeze-out temperature, N π is the number of dark mesons, or dim(G f /H), and t is a factor determined by group theory, ∼ N 5 f for large N f . As a result of freezeout, the number density of dark matter is given by n DM = cT eq s/m π , where T eq = 0.8 eV is the matter-radiation equality temperature, s = (2π 2 /45)g * S (T )T 3 is the entropy density of relativistic particles in thermal equilibrium, and c ≃ 0.54 is the numerical constant. For σv 2 3→2 ≡ α 3 eff /m 5 DM , the freeze-out condition for the 3 → 2 annihilation, Ref. [7] and Ref. [10].
, determines dark matter mass in terms of the effective DM self-coupling.
Thus, for α eff = 1 − 10, we get m DM = 35 − 350 MeV. As will be discussed in a later section, in order to keep dark matter in thermal equilibrium with heat bath, it is necessary to introduce the inter-sector interaction between dark and SM sectors.
On the other hand, the leading 2 → 2 self-scattering comes from the kinetic term (F 2 /16)Tr(∂ µ U∂ µ U −1 ), whose cross section is given by where a 2 is another group theory factor ∼ N 4 f for large N f . The self-interaction cross section is constrained to be σ self /m π 1cm 2 /g. This condition, together with the perturbativity bound of chiral perturbation theory, m π /F < 2π, imposes the dark meson mass to be in the 0.1 − 1 GeV range, depending on the confining gauge group. The group theory factors for possible gauge and flavor symmetries with nonzero WZW terms are summarized in Table I. can be forbidden by appropriate U(1) D charge assignments, such as universal charges up to sign [15]. Even in this case, we cannot prohibit a dark meson self-annihilation in the form of ππ → πγ D through AAAV anomalies 1 [16]. For the 3 → 2 annihilation to be a dominant process for freeze-out, the dark gauge coupling must be extremely small so the gauge kinetic mixing does not give an enough scattering cross section of dark mesons off the SM particles at freeze-out.
For SO(N c ) and Sp(N c ) gauge groups, on the other hand, quarks in the fundamental representation, belong to real and pseudo-real representations, respectively, so there is no distinction between quarks and anti-quarks. As a result, only the Majorana mass terms are allowed. Denoting Weyl spinor indices as α, β, · · · , gauge multiplet indices as r, s, · · · , and flavor indices as i, j, · · · , dark quark mass terms appear as Then, dark mesons are written as an SU (3) Then, the kinetic term (F 2 /16)Tr(D µ UD µ U −1 ), where the covariant derivative is D µ U = ∂ µ U + ig D [Q D , U]V µ , with g D being the dark gauge coupling (or α D ≡ g 2 D /4π being the dark structure constant), provides the leading interactions between dark mesons and dark photon V µ , A remark on the effect of dark photon couplings on the mass splitting is in order. The mass contribution is small for a perturbatively small α D . Since the SIMP scenario works for m π /F 4 [7], for α D = 1/4π, the dark photon contribution to the mass splitting is as small as ∆m 2 π α D Λ 4 /F 2 ∼ α D F 2 ∼ O(10 −2 )m 2 π , i.e. less than 10%. Therefore, the dark meson mass degeneracy is a good approximation. Henceforth, we assume the U(1) D charges given in Eq. (7).

IV. SIMP DARK MESONS WITH SU(N c ) CONFINING GROUP
We consider a gauge kinetic mixing between the U(1) D gauge boson (V µ ) and the U(1) Y gauge boson (B µ ), given by After diagonalizing gauge kinetic and mass terms by where three mass eigenstates (A µ , Z µ , A ′ µ ) are interpreted as photon, Z-boson, and dark photon, respectively, with the masses of the latter two being We get m 2 + ≃ m 2 Z and m 2 − ≃ m 2 V in the χ → 0 limit. To these gauge bosons, electromagnetic (EM) current J EM , neutral Z−current J µ Z , and dark sector current J µ D couple, as The leading interaction between dark and SM sectors is the dark photon coupling to EM current with shifted charges by (c W c ζ t χ )A ′ µ J µ EM . The kinetic mixing parameter, ǫ γ ≡ c W c ζ t χ , and the dark photon mass, m V , are constrained by various experiments.
Dark photon also couples to Z−current. For m V ≪ m Z , however, the Z−current coupling does not give a significant contribution, since the mixing angle approximated by ζ ≃ −s W χ makes the coefficient for dark photon coupling to Z−current, ǫ Z ≡ s ζ + s W t χ c ζ , vanish at the leading order. On the other hand, for the dark photon mass being around the Z−boson mass, the mixing angle gets larger as ζ ≃ (m 2 Z t W ǫ γ )/(m 2 V − m 2 Z ), so we cannot ignore it any longer and the U(1) D gauge boson is interpreted as a 'dark Z boson' [24]. This makes the lower bound on ǫ γ less stringent.
We are now in a position to consider the conditions on m V and ǫ γ that are consistent with the SIMP mechanism. At freeze-out temperature, T F ≃ m π /20 ∼ (5 − 50)MeV, photon, electron/positron, and neutrinos are relativistic particles in thermal equilibrium, and muon and pion begin to be non-relativistic. At that moment, the 3 → 2 self-annihilation from the WZW term is dominant over the other possible annihilation processes whereas dark meson-SM particle scattering processes do not decouple yet, provided that n DM σv ann < n 2 DM σv 3→2 < n SM σv 2 scatt , at the freeze-out temperature, where the number density of a bosonic or fermionic SM particle is given by In order to prevent a pair of dark mesons from annihilating into a pair of dark photons due to the gauge interactions with ππA µ A µ , we require m V > m π . In this region, the ππ → SM SM annihilation rate, estimated as n DM × [O(10 2 )αα D ǫ 2 γ /(N π m 4 V )], is smaller than the 3 → 2 annihilation rate, if For typical parameters such as T F ≃ m π /20, F ≃ m π /5 and α D ≃ 1/4π, the above condition is fulfilled for any value of ǫ γ satisfying the upper bounds given by ground-based experiments.
The only exception appears around m V = 2m π , where ππ → SM SM annihilation rate is improved due to a resonance from 1/(4m 2 π − m V ) 2 . As for the dark meson scattering off the SM particles, the dominant process is π + e ± → π+e ± through the t−channel process, whereas π+γ → π+γ is suppressed by a double kinetic mixing. Ignoring the lepton masses, we obtain the scattering cross section for π + ℓ → π + ℓ averaged over the number of dark mesons N π as where a factor 4 represents the degrees of freedom of U(1) D charged mesons, K ± and π ± . On the other hand, the scattering cross sections of dark mesons off the neutrinos, π + ν → π + ν and the SM pion, π + π ± SM → π + π ± SM , are given by From the condition, n 2 DM σv 3→2 < n SM σv scatt = ℓ=e,µ n ℓ σv scatt,ℓ + n π SM σv scatt,π , Three figures correspond to G c =SU(4), SU(6), and SU(10), respectively. Imposed constraints, distinguishable by colors, are written explicitly, while the allowed parameter space is uncolored.
For m V > 2m π , BaBar and LHC bounds are rescaled taking γ D → 2π invisible decay into account.
the larger m V , the stronger the lower bound on ǫ γ gets, according to that dark meson masses are assumed to be degenerate. For N c = 4, the minimal N c that the SIMP mechanism works, only m π ≃ 0.45 GeV is allowed because the perturbativity condition, x ≡ m π /F < 2π, and the self-interaction bound, σ self /m π 1cm 2 /g, almost coincide [7]. For N c = 6 and N c = 10, a wider range of dark meson masses are allowed such as 0.37 GeV < m π < 0.56 GeV and 0.26 GeV < m π < 0.8 GeV, respectively, and x is fixed by the DM relic density. In both cases, the upper bounds satisfy x = 2π and the lower bounds satisfy x = 5.48 and 4.6, respectively.
We note that dark photon is taken to be heavier than dark mesons in order for the 3 → 2 annihilation to dominate over the ππ → γ D γ D annihilation. As a result, there appears a lower bound, ǫ γ 10 −7 at m V = m π , due to Eq. (21). The lower bound on ǫ γ for a given m V follows from the estimation given by Eq. (21), and the upper bound comes from ground-based experiments. We also find from Eq. (21) that the SIMP condition requires a large kinetic mixing for a large dark photon mass, eventually constrained by ground-based experiments. In summary, for N c < 10, dark photon masses are allowed up to m V ∼ 10 3 GeV with varying limits on ǫ γ . We note that there are more allowed values of ǫ γ around m V ≃ m Z due to the non-negligible contribution from ǫ Z . The bounds get stronger near m V = 2m π , where the dark meson annihilation into a pair of SM particles becomes enhanced as discussed previously. confining group for them, we showed that the combination of the SIMP conditions with various ground-based experiments searching for dark photon can restrict the parameter space to m π < m V 10 3 GeV and 10 −7 < ǫ γ < (10 −3 − 10 −2 ), for dark gauge coupling of order one and N c < 10.