Graviton energy spectra arising from the KSVZ axion model

Axion, the goldstone boson arising from the spontaneous breaking of a global $U(1)$ Peccei-Quinn symmetry, provides a dynamical solution to the strong CP problem and is an excellent dark matter candidate. Various experiments are designed to search for the axion, however no confirmative signal has been observed. On the other hand, there are also hypothetical heavy particles in axion models, such as the heavy scalar $s$, which is the CP-even component of the complex scalar that carries $U(1)_{PQ}$ charge, and the vector-like heavy quark (VLQ) in the Kim-Shifman-Vainshtein-Zakharov~(KSVZ) axion model. Studying signals induced by them are helpful for axion searches. In this paper, we calculate the graviton bremsstrahlung energy spectrum arising from the decay of the heavy scalar or VLQ in the KSVZ model. The result shows that these heavy particles can emit ultrahigh-frequency gravitational waves (GWs), with the peak frequency depending on the model’s parameter inputs. In addition, the graviton spectrum is distinguished from the thermal GW background at high frequencies if there is an early matter-dominated era induced by these heavy particles. Future measurements of ultrahigh-frequency GWs may provide indirect evidence for the KSVZ axion.

💡 Research Summary

The paper investigates a novel source of ultra‑high‑frequency gravitational waves (GWs) within the Kim‑Shifman‑Vainshtein‑Zakharov (KSVZ) axion framework. In addition to the familiar QCD axion, the KSVZ construction introduces a heavy complex scalar Φ whose radial excitation (the “saxion” s) and a vector‑like quark (VLQ) Q acquire masses after the Peccei‑Quinn (PQ) symmetry breaks at a scale f_a. The authors focus on the regime where the PQ breaking occurs after reheating (post‑inflationary scenario), so that s and Q are initially in thermal equilibrium with the Standard Model plasma.

First, the authors write down the relevant Lagrangian terms, showing that s couples to a pair of VLQs via a Yukawa coupling y_Q and to the Higgs doublet via a quartic coupling λ_ΦH. The masses are m_s = √2 λ_Φ f_a and m_Q = y_Q f_a/√2. Decay widths for the channels s → Q Q̄, s → HH, and Q → H q (or other SM quarks) are derived (Eqs. 9‑11). By imposing that the VLQ lifetime be short enough to avoid cosmological problems (τ_Q ≲ 10⁻² s) yet long enough to generate a temporary matter‑dominated era, the authors obtain stringent upper bounds on y_Q and y_{Qq} (∼10⁻³–10⁻⁵ for a typical f_a ≈ 10¹² GeV).

The cosmological evolution is then solved analytically. The energy densities of s and Q obey Boltzmann equations with Hubble dilution and decay terms (Eqs. 14‑17). For sufficiently small couplings, the universe experiences two distinct matter‑dominated intervals: one when s dominates before it decays into VLQs, and a second when the non‑relativistic VLQs dominate before they themselves decay. Figure 1 illustrates this two‑step evolution for a benchmark point (m_s = f_a = 10¹² GeV, y_Q = 10⁻⁶, y_{Qq} = 10⁻⁷). The matter‑dominated phases are crucial because they enhance the production of gravitons via bremsstrahlung relative to the standard thermal background.

The core of the work is the calculation of the graviton bremsstrahlung spectrum generated during the 2‑body decays s → Q Q̄ g and s → HH g (and similarly for Q decays). Using the standard graviton‑matter interaction vertices, the authors adopt the squared matrix element from the literature (Eq. 25) and integrate it over the three‑body phase space with the appropriate Bose‑Einstein/Fermi‑Dirac distributions (Eqs. 22‑24). The collision term C in the Boltzmann equation (21) yields a differential energy density dρ_GW/dE_g, which after redshifting gives the present‑day spectral density Ω_GW(f) (Eq. 26). The integration limits are expressed in terms of dimensionless variables x_Q = 2E_Q/m_s, x_g = 2E_g/m_s, and z_Q = 2m_Q/m_s, ensuring the correct kinematic boundaries.

Numerical evaluation of the spectrum shows a pronounced peak at frequencies f ≳ 10¹⁰–10¹² Hz, depending on the chosen parameters. The peak amplitude can exceed the stochastic thermal GW background (Ω_GW ∼ 10⁻¹⁶) by several orders of magnitude, especially when a matter‑dominated era is present. The peak position scales roughly as f_peak ∼ (m_s/10¹² GeV) × 10¹² Hz, while its height is proportional to (y_Q² / m_s²) and inversely to the Planck mass squared, reflecting the gravitational coupling suppression.

The authors argue that such ultra‑high‑frequency GWs are potentially observable with future experimental concepts, such as resonant microwave cavities, optomechanical sensors, or superconducting circuits designed for GHz–THz gravitational wave detection. Detection of a spectral shape matching the predicted bremsstrahlung profile would provide indirect evidence for the existence of heavy KSVZ states and, by extension, support the axion solution to the strong CP problem.

In conclusion, the paper establishes a concrete link between heavy particle physics in the KSVZ axion model and a distinctive high‑frequency gravitational wave signature. It highlights how the lifetime and coupling choices of s and Q control both the cosmological matter‑dominated epochs and the resulting GW spectrum. This work opens a new observational window—ultra‑high‑frequency gravitational wave astronomy—complementary to traditional axion searches (haloscopes, helioscopes, EDM experiments) and offers a pathway to probe otherwise inaccessible sectors of beyond‑Standard‑Model physics.