Three-photon and three-gluon annihilation decays of vector Quarkonia in a unified instantaneous Bethe–Salpeter framework

Abstract

We investigate the electromagnetic and hadronic annihilation decay widths of S -wave vector quarkonia, namely charmonium (

) and bottomonium (

$$\Upsilon (nS)$$ Υ ( n S )

) with radial excitations

$$n=1$$ n = 1

–4, within the framework of the instantaneous Bethe–Salpeter (BS) equation. The interaction kernel combines a harmonic confinement potential with one-gluon exchange, using a single unified parameter set for both

$$c\bar{c}$$ c c ¯

and

$$b\bar{b}$$ b b ¯

systems. The obtained relativistic wave functions are used to compute the decay widths for the processes of

$$nJ/\psi \rightarrow 3\gamma $$ n J / ψ → 3 γ

,

$$nJ/\psi \rightarrow 3g$$ n J / ψ → 3 g

,

$$n\Upsilon \rightarrow 3\gamma $$ n Υ → 3 γ

, and

$$n\Upsilon \rightarrow 3g$$ n Υ → 3 g

. For the ground state,

$$\Gamma _{J/\psi (1S)\rightarrow 3\gamma }=1.004\times 10^{-3}\,\text {keV}$$ Γ J / ψ ( 1 S ) → 3 γ = 1.004 × 10 - 3 keV

agrees well with experiment, while

$$\Gamma _{J/\psi (1S)\rightarrow 3g}=59.28\,\text {keV}$$ Γ J / ψ ( 1 S ) → 3 g = 59.28 keV

and

$$\Gamma _{\Upsilon (1S)\rightarrow 3g}=43.42\,\text {keV}$$ Γ Υ ( 1 S ) → 3 g = 43.42 keV

closely reproduce measured values. The decay widths decrease with radial excitation due to node effects in the wave functions. The ratios

$$\Gamma _{3g}/\Gamma _{3\gamma }$$ Γ 3 g / Γ 3 γ

are found to be

$$0.59\times 10^{5}$$ 0.59 × 10 5

for

$$J/\psi $$ J / ψ

and

$$0.87\times 10^{6}$$ 0.87 × 10 6

for

$$\Upsilon $$ Υ

, consistent with the expected scaling

$$\Gamma (3g)/\Gamma (3\gamma )\propto (\alpha _s/\alpha _{\textrm{em}})^3$$ Γ ( 3 g ) / Γ ( 3 γ ) ∝ ( α s / α em ) 3

. These results demonstrate that the Bethe–Salpeter approach with a unified parameter set provides a reliable relativistic framework for quarkonium annihilation processes.