Researches on Dark Matter Using e⁺ e⁻ Collider

Type 1. Dark Photon that Decays to Dark Matter case, nothing is left in the detector.

Fig. 5 shows the plane of M_A' vs. M_χ with different characteristics of the single-photon and missing E_T signals. A' decays to the dark matter on-shell or off-shell with different gamma spectrum. The results are from the radiative production in an collision. In the final state, there exists only single-photon with the energy, $E_{\gamma }^{*}=\left(S-M_{A^{\prime }}^2\right)/2\sqrt{S.}$ Here, s is the square of CM energy at the e⁺e⁻ collider and is dark photon mass (Essig et al. 2013). This requires a high single-photon trigger. However, it is not available in the Belle experiment, while the BaBar experiment had been implemented with a single-photon trigger (BABAR Collaboration 2008). It will be implemented in the Belle II experiment.

Fig. 5. The M_A' vs. M_χ plane. There are four regions: off-shell heavy A' region. on-shell decay to dark matter regions, off-shell light A' region and, long-lived ultra-light A' region (Essig et al. 2013).

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As shown in Fig. 6, dark photon A' can decay directly to the SM (Batell et al. 2009). Fig .7 shows branching ratios of each decay channel (Batell et al. 2009).

Fig. 6. Feynman diagram showing the dark photon decaying to the SM particles.

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Fig 7. Branching ratios of each decay channel. V → e⁺e⁻ (dashed), V → μ⁺μ⁻ (dotted), V → τ⁺ τ⁻ (dotted-dashed), and V → hadrons (solid) (figure in ref. Kacurova (2009)).

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The cross section of dark photon $A^{\prime }\left(\sigma \infty \frac{{\epsilon }^2}{E_{cm}^2}\right)$ is inversely proportional to the square of the CM energy (See the left Feynman diagram of Fig. 3). Therefore, it requires low energy (1–10 GeV) with a high-intensity collider (BaBar, Belle and KLOE). A broad direction search is required and underway.

Fig. 8 shows the Feynman diagram of the Higgsstrahlung. Here, h' decays depending on Mh' and M_A'. We can construct a two-dimensional plot for Mh' and M_A' to measure the coupling constant between the dark photon to the dark Higgs, α_D. As shown in Fig. 9, three cases exist:

Fig. 8. Feynman diagram of Higgsstrahlung.

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Fig. 9. The plane of M_h' vs. M_A' plot. Three cases exist depending on the M_A' mass.

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Case (1): M_h' > 2 M_A' : h' → A'A', very low background

Case (2): M_A' < M_h' < 2 M_A' : h' → A'A'*

Case (3): M_h' < M_A' : h' is long lived and h' → l+ l−, π⁺ π−

This is rather similar to that of type 2. If the SM fields are under a dark force and charged, the lepton universality may be broken. A' may decay only to heavy leptons (Shuve & Yavin 2014). The theory can explain the muon anomalous magnetic moment. Α' → υν̅ if A' couples. Fig. 10 shows the Feynman diagram of e⁺e⁻ → μ⁺μ⁻A' and A' → μ⁺μ⁻ decay channels.

Fig. 10. Feynman diagram showing A' coupling only to heavy leptons.

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The first search is the dark photon (ηe⁺e⁻), as follows (KLOE-2 Collaboration 2016a):

Fig. 11 shows the Feynman diagram with the final state of ηe⁺e⁻.

Fig. 11. Feynman diagram showing the final state of ηe⁺e⁻.

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The second search is the dark photon (e⁺e⁻γ & μ⁺μ⁻γ). Fig. 12 shows the final state Feynman diagram of e⁺e⁻γ or μ⁺μ⁻γ (KLOE-2 Collaboration 2016a).

Fig. 12. The final state Feynman diagram of e⁺e⁻ γ or μ⁺μ⁻ γ.

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KLOE investigates in the dark photon mass range below 1 GeV from the motivation of light gauge boson. The final state is π⁺π-γ or μ⁺μ-γ. No invisible decay is assumed due to M_A' < M_χ. The evidence for a signal does not exist. KLOE set a limit at 90 % confidence level (CL) on ε² in the M_A' range of 527- 987 MeV, which is more constraint than the previous ones beyond 700 MeV (KLOE-2 Collaboration 2016b).

As shown in Fig. 13, the final state is μ⁺μ⁻ and missing ET (h'). This is case (3) Higgssrahlung described in section 2.1.3. The h' long lived and invisible. The evidence of signal does not exist. KLOE set a limit on 90 % CL to its parameters in the range 2Mμ < M_A' < 1,000 MeV, Mh' < M_A' (KLOE-2 Collaboration 2015).

Fig. 13. feynman diagram showing the Higgsstrahlung at KLOE experiment.

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Using 2.93 fb⁻¹ dataset, BES III searches for a dark photon. BES III examines the initial state radiation reactions as follows:

In this search, the invariant mass distribution of lepton pairs appears an enhancement as dark photon. The evidence of signal does not exist.in the mass range of (1.5 - 3.4) GeV/c². BES III set a limit on 90 % on the mixing strength, ε² (BES III Collaboration 2017).

The BaBar experiment has one photon + with missing E_T trigger. Therefore, this experiment measures the decay channel of

The first search was Υ(3S) →γA' and A' → invisible (BABAR Collaboration 2008). Through the next-to-minimal supersymmetric SM (NNSSM), such an object appears. BaBar investigate the events of single-photon with large missing ET by tagging the two-body decay of Υ(3S). The evidence of signal does not exist for such processes in 122 × 10⁶ Υ(3S) events. BaBar set a upper limit on 90 % CL on the branching ratio of B(Υ(3S) → γ A') × B(A' → invisible) at (0.7−31) × 10⁶ in the dark photon mass range M_A' ≤ 7.8 GeV (BABAR Collaboration 2008).

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The second search was Υ(2S) → π⁺π⁻ Υ(1S), Υ(1S)→γA⁰, where A⁰ decays invisibly. A⁰ is invisible. It is either a light Higgs boson with zero spin, or a dark matter pair with zero spin. BaBar investigate the events of single-photon with large missing ET by Y(1S) tagging (BABAR Collaboration 2011). The evidence of signal for such processes does not exist in the range MA⁰ ≤ 9.2 GeV/c² and M_χ ≤ 4.5 GeV/c² using 98 × 10⁶ Υ(2S) events. BaBar set an upper limit on the model which has light dark matter. The results are:

at 90 % CL upper limits (BABAR Collaboration 2011).

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BaBar searches for e⁺e⁻ → Υ(2S, 3S), Υ(2S, 3S)→ π⁺π⁻ Υ(1S), Υ(1S)→γA⁰ and A⁰ → μ⁺μ⁻ (BABAR Collaboration 2013). BaBar selects Y(1S) events by tagging the pion pair. The evidence of signal does not exist for the A⁰ production examining 116.8 × 10⁶ Υ(3S) events and 92.8 × 10⁶ Υ(2S) events. BaBar set an upper limit on 90 % CL on the product branching ratio as follows:

Using 516 fb⁻¹ dataset, BaBar experiment searches for the dark Higgs boson and set the upper limits (BABAR Collaboration 2012). This is case (1) Higgsstrahlung in Section 2.1.2. The experiment set a limit on 90 % CL on the crosssection on e⁺e⁻ → hA' and h' → A'A depending on h' and A' masses. The experiment uses the conservative approach with no background subtraction (BABAR Collaboration 2012).

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The channel is e⁺e⁻ → μ⁺μ⁻ Z' and Z' → μ⁺μ⁻. The Z' lepton coupled the dark photon mode. Using 514 fb⁻¹ dataset, BaBar searches for the Z' which couples only to heavy leptons. The evidence of signal does not exist for the Z' mass range of (0.212 − 10) GeV. BaBar set the lower limit on the coupling parameter g' as 7 × 10−4, which improves the bounds compared to neutrino experiments (BABAR Collaboration 2016).

Type 1 requires a high single-photon trigger. This is not available in the Belle experiment while BaBar was implemented with a single-photon trigger (BABAR Collaboration 2008).

Dark photon decays to the SM particles such as e⁺e⁻, μ⁺μ⁻ and π⁺π⁻ (Kacurova 2009). Here, the dark photon mass range is between (0.1⁻¹0 ) GeV/c². This analysis includes the dark photon decays from prompt or a displaced vertex. Fig. 14 shows the decay channel with the displaced vertex. This is also called the “long-lived gauge boson” instead of dark photon. This work is underway. Belle also investigate a dark vector gauge boson which decays to π⁺π⁻ by tagging D*0 decays (Belle Collaboration 2016). The evidence of signal does not exist. Belle set a mass-dependent limit on the baryonic fine structure constant of 10⁻³ − 10⁻² in the dark photon mass range of 290-520 MeV/c² (Belle Collaboration 2016).

Fig. 14. Dark photon to the SM particles.

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The Belle experiment searches for the Higgsstrahlung case (1) described in section 2.1.3. It has very low background by prompting h' and A'. The Belle measures exclusively three pairs of charged track with the same invariant mass and total energy of event, inclusively two pairs of charged track with the same invariant mass, and the third A' from the four momenta of the e⁺e⁻ collider beam constraint system. In conclusion, Belle used 10 exclusive channels: 3(l⁺l⁻), 2(l⁺l⁻) (π⁺π⁻), 2(π⁺π⁻) (l⁺l⁻), 3(π⁺π⁻) with l = e, μ, and three inclusive channels: 2(l⁺l⁻)X with X dark photon (missing mass). The evidence of signal does not exist. Belle set a limit on 90 % CL on the mixing strength (Belle Collaboration 2015).

For the one photon + with missing ET for the channel, a single-photon trigger is required. Belle II develops it.

The resolution of invariant mass of μ⁺μ⁻ at the Belle II experiment is much better than that of Belle experiment. Therefore, Belle II has much better efficiency for these channels (Alexander et al. 2016).

The Belle II experiment also proposed to search axionlike particles (ALP), as shown in Fig. 15. They are pseudoscalars and couple to bosons. Unlike QCD axions, ALPs have no relation between mass and coupling, assuming all axions decay into two photons a → γγ. As shown in Fig. 16, four regions exist on the plane of g_Aγγ−m_A: the invisible, displaced, merged, and resolved. For the invisible region, the ALP decays outside of the detector or into the invisible particles: the single-photon final state. For the displaced region, the searches for invisible and visible APL decays into this region. For the merged region, two of the photons overlap or merge. For the resolved region, three resolved high energetic photons (Mimasu & Sanz 2014).

Fig. 15. Feynman diagram of axion-like particles.

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Fig. 16. The plane of g_aγγ−m_a for the axion-like particle (Mamisu & Sanz 2014).

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