Dark light shining on $B o K^{(*)} E_{ m miss}$

Recent Belle II data on $B^+ \to K^+ E_{\rm miss}$ show an excess consistent with a two-body decay involving a light invisible particle with mass around $2.1,\mathrm{GeV}$. We present a UV-complete explanation based on a Higgsed $U(1)’$ gauge symmetry with a light vector boson $Z’$ and a vector-like top partner, which naturally enhances $b \to s$ transitions. While the minimal model can reproduce the required $B \to K^{(*)} Z’$ rate, it is excluded by LHCb searches for resonant dimuon decays due to unavoidable loop-induced couplings of $Z’$ to charged leptons. We show that a minimal extension with an additional light $U(1)’$-charged singlet fermion allows $Z’$ to decay dominantly invisibly, evades existing constraints coming also from dark photon and collider searches as well as Higgs measurements, and can simultaneously account for the Belle II excess and the observed dark matter abundance through resonant thermal freeze-out.

💡 Research Summary

The paper addresses the 2.9 σ excess reported by Belle II in the decay B⁺→K⁺ + missing energy, which is compatible with a two‑body decay B⁺→K⁺ X where X is a light invisible particle of mass ≈ 2.1 GeV. The authors construct a UV‑complete framework based on a Higgsed U(1)′ gauge symmetry. The field content includes a vector‑like top partner T′ charged under both the SM and U(1)′, and a complex scalar Φ that acquires a vacuum expectation value, breaking U(1)′ and giving mass to a new gauge boson Z′ (M_{Z′}≈2.1 GeV). Mixing between the SM top and T′, controlled by Yukawa couplings y_t, y_{tT} and the gauge coupling ˜g q′, induces flavour‑changing b→s transitions that can generate B→K^{(*)} Z′ at the rate required by the Belle II data.

At one loop the heavy quarks generate kinetic mixing (ε_A, ε_Z) and mass mixing (δ_Z) between Z′ and the SM photon/Z. These mixings inevitably give Z′ couplings to charged leptons proportional to the SM Z or photon currents. Consequently, Z′ would decay visibly into μ⁺μ⁻ (or e⁺e⁻) and would be seen as a resonant dimuon signal in B→K^{(*)} μ⁺μ⁻. Existing LHCb searches for such resonances exclude the minimal model for any realistic choice of parameters, even when kinetic mixing is tuned to cancel.

To evade this constraint the authors extend the model by adding a light SM‑singlet fermion χ that carries the same U(1)′ charge. If m_χ < M_{Z′}/2, Z′ decays dominantly invisibly as Z′→χ χ̄, suppressing the dimuon branching ratio below experimental limits. The χ particle is stable due to the unbroken Z₂ parity and thus becomes a dark‑matter candidate. The authors show that for couplings ˜g q′≈10^{‑3}–10^{‑2} and y_{tT}≈0.1–0.3, χ annihilates via resonant s‑channel Z′ exchange (χ χ̄→Z′→SM) with the correct thermal relic abundance (Ω_{DM}h²≈0.12). The required parameter space simultaneously satisfies:

• Flavour constraints: B→K^{(*)} νν̄, D⁰→π⁰ νν̄, K→π νν̄ limits are respected because the Z′ coupling to neutrinos is suppressed by the small mixing angles.
• Electroweak precision tests: The induced δ_Z and ε_Z lead to shifts in the S and T parameters well within the current 95 % CL.
• Dark‑photon searches: With ε_A≲10^{‑4} the model evades bounds from NA64, BaBar, Belle II dark‑photon analyses.
• Higgs physics: The scalar portal coupling λ′ induces a small mixing between the SM Higgs and the Φ scalar; the mixing angle is kept below ≈ 0.1 to satisfy Higgs signal‑strength measurements.
• Collider limits: Direct Z′ production with visible decays is negligible; displaced‑vertex searches are avoided because Z′ decays promptly to χ χ̄.
• Direct and indirect DM detection: χ‑nucleon scattering proceeds via loop‑suppressed Z′ exchange, yielding cross sections below current XENONnT/LZ limits; annihilation signals are p‑wave suppressed today, evading gamma‑ray constraints.

The paper provides detailed analytical expressions for the mass matrices, mixing angles, and loop‑induced kinetic/mass mixing, and presents numerical scans illustrating viable regions. It also discusses possible future probes: improved Belle II measurements of B→K + invisible, LHCb searches for B→K^{(*)}+missing energy, and dedicated low‑mass dark‑photon experiments could test the kinetic‑mixing parameter space; collider searches for the vector‑like top partner T′ (mass ≳ 1 TeV) and for the scalar Φ (mass ≳ few hundred GeV) offer complementary tests.

In summary, the authors demonstrate that the minimal Higgsed U(1)′ model is ruled out by LHCb dimuon searches, but a minimal extension with a light U(1)′‑charged fermion χ restores viability. This extended framework simultaneously explains the Belle II B→K + missing‑energy excess and provides a thermal dark‑matter candidate, while remaining consistent with a broad array of flavour, electroweak, collider, and astrophysical constraints. The model makes concrete predictions that can be probed in upcoming Belle II, LHCb, and dark‑photon experiments.