The “pions” have self-interactions whose cross sections are determined by the size of “pions,” which is of order Λ-1. Representing an O(1) factor by c1, we write the cross section as σelamπ=(4π)4c12mπ4πΛ4≃2.7  cm2/g(c1cΛ2(4π)-1)2(mπ100  MeV)-3(1)from the dimensional analysis.²

We assume c1cΛ2≲(4π)-1 throughout this paper so that the scattering cross section is less than the geometrical cross section, 4π/mπ2, that is below the unitarity bound for v<c [38].

There must be a nonzero kinetic mixing ε between the U(1)d gauge boson and the U(1)Y gauge boson because it is allowed by any symmetry [36]. There are two types of kinetic mixing terms in theories consisting simultaneously of both a “monopole” and an “electron”: ε′BμνFμν and εBμνF˜μν, where Bμν and Fμν are the field strengths of U(1)Y and U(1)d gauge bosons, respectively, and F˜μν≡(1/2)ϵμνρσFρσ. If the CP symmetry is conserved, either of these mixing terms is allowed.³

One may think that εBμνF˜μν itself violates the CP symmetry. In general, either of Fμν and F˜μν can be chosen to be a tensor, and the other one is a pseudotensor. If we choose the definition in which Bμν and F˜μν are tensors and B˜μν and Fμν are pseudotensors, the kinetic mixing term εBμνF˜μν conserves the CP symmetry. In this case, dark “pions” transform as π→-π (rather than π→-πT) under the CP, so that [Tr[π∂μπ∂νπ]-(μ↔ν)] is also a tensor and can be mixed with F˜μν.

The U(1)d gauge theory may be conformal in the presence of “monopole” as well as “electrons” [27,28], which implies that the gauge field strength Fμν has an scaling dimension larger than 2 as is guaranteed by the unitarity bound [45]. As a result, the kinetic mixing terms are irrelevant operators and are suppressed at low energy [21], if present. This naturally results in small ε′ and ε in our model. Hereafter, we represent Bμν as the photon field strength and absorbs the Weinberg angle into ε′ and ε for notational simplicity.

In this paper, we mainly consider the case with εBμνF˜μν and without ε′BμνFμν for simplicity unless otherwise stated. In the dual basis, our model looks similar to the standard spontaneously broken U(1)d gauge theory, where the U(1)d symmetry is (spontaneously) broken by the condensation of the “Higgs” field (i.e., the scalar “monopole” in the original basis) and the kinetic mixing term looks the same as the usual one, εBμνFμν. Then we can quote constraints on the kinetic mixing parameter to compare our result with the present and future constraints. We will explain the case only with ε′, which leads to a similar result to the case only with ε.

Here, we note that F˜μν does not satisfy the Bianchi identity, ϵμνρσ∂νF˜ρσ=0, in theories consisting simultaneously of both a “monopole” and an “electron” (see, e.g., Ref. [46]). Then an operator mixing between F˜μν and Tr[π∂μπ∂νπ] is allowed in those theories. Therefore, once we allow the nonzero kinetic mixing, εBμνF˜μν, we can have a term like L⊃cε(4π)2Λ3εBμνTr[π∂μπ∂νπ],(2)where cε is an O(1) constant. This operator leads to a semiannihilation process of ππ→πγ only in the presence of a “monopole” and “electrons.” If F˜μν satisfied the Bianchi identity, one could write F˜μν=∂μV˜ν-∂νV˜μ with V˜μ being a (magnetic) gauge field of U(1)d. Then the kinetic mixing operator BμνF˜μν could be written as -2∂μBμνV˜ν=0 after the integration by parts for the on shell photon. However, F˜μν does not satisfy the Bianchi identity in the presence of a “monopole” as well as “electrons”. There is no reason that we prohibit the operator of Eq. (2) and the on shell photon is produced by the annihilation process, ππ→πγ.

The operator of Eq. (2) vanishes for Nπ<3, since it is antisymmetric in the flavor SU(NF), so that we assume NF≥2 in our model. We note that the “pions” transform as an adjoint representation of the flavor SU(NF). The two “pions” in the initial state must be antisymmetric in the flavor SU(NF) to contact with the one “pion” in the final state. On the other hand, the initial state of the semiannihilation process must be symmetric in terms of the “pion” exchange because “pions” are bosons. These observations imply that the initial angular momentum must be antisymmetric and the semiannihilation process is p wave suppressed. We thus expect that its cross section can be estimated as ⟨σv⟩ππ→πγ∼cε2ε2(4π)4mπ44πΛ6(Tmπ),(3)where a factor of cε2ε2(4π)4/Λ6 comes from Eq. (2) and the power of mπ is determined by the dimensional analysis. We absorb an O(1) uncertainty into cε. This interaction is in thermal equilibrium at a temperature higher than mπ for cεε≳4×10-12cΛ-3(mπ/100  MeV)1/2. The temperature of the “pions” is the same as that of the SM sector until the semiannihilation process freezes out at T/mπ∼1/20.

As the temperature becomes lower than the “pion” mass, the number density of “pions” is suppressed by the Boltzmann factor and eventually the ππ→πγ semiannihilation process freezes out. We note that the ππ→πγ semiannihilation process is similar to but is slightly different from the standard annihilation process in the weakly interacting massive particle (WIMP) scenario. The important difference is that the “pion” in the final state can be relativistic and may heat the dark sector [14,47,48]. From the Boltzmann equation of the “pions,” the evolution equations of the yield Yπ (≡nπ/s) and the inverse temperature xπ (≡mπ/Tπ) are approximated as ddxYπ≈-λx2Yπ2,(4)xddx(xπx)≈xπx+23λ¯Yπ(xπx)2,(5)for x (≡mπ/T) >xFO (≡mπ/TFO), where s=(2π2/45)g*T3, T is the temperature of the SM particles, and TFO is the freeze-out temperature (see Ref. [47] for the original equations without using approximations). The effective number of relativistic degrees of freedom, g*, is taken to be about 10. The dimensionless reaction rates are given by λ=xs⟨σvrel⟩2H,λ¯≈-(γ-1)λ,(6)where γ (=5/4) is the Lorentz factor that DM achieves through semiannihilation.

Assuming xFO∼20, we numerically solve Eqs. (4) and (5). The time evolutions of the yield and the temperature of “pions” are shown as black curves in Fig. 1, where the yield is normalized by YπFO≡2xFO/λ(xFO). The red curve in the upper panel is the one without the self-heating while that in the lower panel is xπ=0.033x2/xFO to which the numerical result asymptotically approaches. Thus, we obtain the asymptotic value of the yield and the temperature of “pions” as YπFO≃cY2xFOλ(xFO),xπ≃cxx2xFO,(7)for xπ≫xFO, where cY=O(0.1) and cx=O(0.1) are numerical constants.⁴

The initial condition is taken to be Yπ=cinixFO2/λ(xFO) and xπ=x at x=xFO with cini being an O(1) constant. The numerical coefficients cY and cx depend on cini only logarithmically, while they linearly depend on xFO-1.

The second term in the right-hand side of Eq. (5) becomes negligible after the freeze-out if the semiannihilation process is p wave suppressed and λ¯∝1/xπ. Then, the temperature of the “pions” scales as Tπ∝1/a2 just like the nonrelativistic matter and the DM “pions” are cold, where a is the scale factor. This is in contrast to the case of a s-wave semiannihilation process discussed in Ref. [47], where it is found that Tπ∝1/a because both the first and second terms in the right-hand side of Eq. (5) are relevant and are balanced until the self-interaction freezes out. In the latter case, the temperature of DM is not that small and DM is warm, which is tightly constrained by measurements of the Lyman-α forest [48]. On the other hand, the temperature of DM decreases faster, and DM is cold in our model.

We show the allowed region of the kinetic mixing parameter ε2 in Fig. 2. We assume that c1,cΛ,cε∈(0.1,1) with a condition of c1cΛ2<(4π)-1 (see footnote 3) for a conservative analysis, while we take cY=0.1 and cmV=1/4 for simplicity. The shaded regions are parameters in which the DM relic abundance can be consistent with the observed DM abundance and the self-interaction cross section can be σela/mπ∈(0.1,1)  cm2/g. In the darkly shaded region, σela/mπ can be as large as 1  cm2/g, while in the lightly shaded region it is smaller than 1  cm2/g but can be larger than 0.1  cm2/g. The upper-left corner of the shaded region is bounded by the condition that cΛ should not be smaller than about 0.1 in Eq. (8). In the upper-right (lower-left) corner of the figure, the self-interaction cross section of “pions” becomes too small (large) to be consistent with the observations of the small-scale structure. If ε2 is smaller than about 10-11 and Nπ≥5, the 3→2 annihilation process becomes relevant during the freeze-out process as we will see shortly.

Since there is no π-π-γ (or dark photon) interaction due to the flavor SU(NF), the “pions” cannot be detected by the direct-detection experiments of DM. On the other hand, the dark photon can be produced via the kinetic mixing and can be discovered by some experiments employing missing momentum and/or energy techniques. In the figure, we plot the constraints on the kinetic mixing parameter by BABAR [49,50,55] and NA64 [51,56] in the magenta and green lines, respectively. We can see that most of the parameter space is consistent with the present upper bound. The expected sensitivities of future experiments are shown by the dashed lines for Belle II (magenta) [52,53], NA64 (green) [54], LDMX (blue), and Extended LDMX (red) [1] experiments. We find that the (Extended) LDMX experiment as well as Belle II experiment can cover a large parameter space.

Note that the dark photon cannot decay into two “pions” in our model. This implies that the dark photon cannot decay solely into the dark sector for the case of mV<3mπ. On the other hand, the dark photon dominantly decays into the dark “pions” for the case of mV>3mπ. The LDMX experiment is designed to measure missing momentum in this kind of process. As we hope to indirectly detect the DM particle by LDMX-like experiments, we assume mV=4mπ (>3mπ), i.e., cmV=1/4, to plot the figure. We predict that mV is larger than about 30 MeV because we require mV>3mπ and mπ≳10  MeV.

Here, we comment on the case in which there is only the other kinetic mixing term ε′BμνFμν rather than εBμνF˜μν. In this case, Eq. (2) should be replaced by a term like cε′(4π)2/Λ3εB˜μνTr[π∂μπ∂νπ] though our analysis of the semiannihilation process does not change much. The SM charged particles cannot emit on shell dark photons while the dark-sector particles can be produced via the off shell (dark) photons via the kinetic mixing. We expect that the cross section of such a process with missing particles is then given by the replacements of mV by Λ and ε2 by ε′2 with an additional factor of NFαD/(2π)ln(E/Λ) (∼O(1)) for E≳Λ, where E (=O(1)  GeV) is the energy of the scattering process [50].⁵

One may think that the cross section is dominated by a low-energy contribution near the threshold of 3mπ [50]. In our case, however, it is negligible due to the p-wave suppression effect.

We also comment on the region near the lower bound on the “pion” mass (∼10  MeV). As the “pions” are nonrelativistic and are suppressed by the Boltzmann factor during the freeze-out process of neutrinos, the effect of “pion” decoupling is almost negligible for observables such as the effective number of neutrinos. However, it is argued that its effect can be detected in the near future by the Simons Observatory [57] and CMB-S4 [58,59] if the “pion” mass is as small as about 10–15 MeV [60].

Finally, we note that the constraint from the indirect detection experiments of DM is not relevant in our model because the semiannihilation process is p wave suppressed and is not efficient in the galactic scale (see, e.g., Ref. [61]).

The “pions” may experience a 3→2 annihilation process via the following operator: cWZW(4π)3N3/2Λ5ϵμνρσTr[π∂μπ∂νπ∂ρπ∂σπ].(9)This term is allowed by any symmetry and is an analogy to the Wess-Zumino-Witten term in strong SU(N) gauge theories. It trivially vanishes for Nπ<5, namely NF<3. The cross section for the 3→2 annihilation process is calculated as [3] ⟨σv2⟩3→2=(4π)6cWZW23755mπ52πNFΛ10T2mπ2.(10)We should check that it is not efficient during the freeze-out of the 2→2 semiannihilation process induced by Eq. (2). The condition is written as ⟨σv2⟩3→2(nπeq(TFO))2≲⟨σv⟩ππ→πγnπeq(TFO)≃H(TFO).This condition is satisfied when ε≳2×10-6cε-1(cWZW0.1)3/5(mπ100  MeV)1/10,(11)where we consider the case in which the relic abundance of “pions” is consistent with the observed DM abundance. In Fig. 2, the shaded region satisfies this condition with cWZW=0.1 and cε=1. However, we note that Eq. (9) trivially vanishes and the constraint of Eq. (11) is not applied for the case of NF=2 (Nπ=3), which is the minimal case for semiannihilation to work in our model.

We also note that Eq. (2) leads to a 3→2 semiannihilation process [62] such as ππe→πe, where e generically represents the SM charged particles. The cross section of this process is roughly estimated as (σv2)ππe→πe∼(σv)ππ→πγ(4πα)/mπ3. Here, ⟨σv2⟩ππe→πene is suppressed by a factor of order 4παne/mπ3 compared with ⟨σv⟩ππ→πγ. This is as small as 10-6 at the time of freeze-out (ne∼T3) and much smaller by many orders of magnitude at the present epoch, so that the process is not relevant for setting the relic abundance nor leading indirect-detection signals. Since (σv)ππ→πγ∼(v/c)210-9/GeV2, (σv2)ππe→πe is many orders of magnitude smaller than the one predicted in the model of Ref. [62]. Therefore, the process is not relevant for leading direct-detection signals.

There must be a nonzero mixing between the “monopole” ϕ and the SM Higgs field H because the following interaction term is allowed by any symmetry: Vmix=λ|ϕ|2|H|2,(12)where λ is a constant. After the Higgs and “monopole” condensations, the mixing angle between the “monopole” and the SM Higgs field is given by θ≃0.023λcmix(mπ1  GeV)(mV3mπ),(13)where we assume that the “monopole”-condensation scale is related to mπ by an O(1) factor cmix.

There is a strong collider constraint on the mixing parameter from the Higgs-decay channel into two “monopoles” [63]. The “monopoles” can decay into muons after they are produced from the Higgs decay [64]. In this case, the branching ratio of the Higgs decay into the “monopoles” must be smaller than about 1% [65], which requires that the quartic coupling λ must be smaller than of order 10-3. Such a small coupling may be naturally realized in our model because our model may be conformal above the “monopole” and “electron” mass scale, and the “monopole” has a relatively large anomalous dimension [27,28]. The search for the Higgs decay into muons may also be an interesting direction to test our model in the near future.