This paper builds a simple and testable bridge between theory and lattice simulations for a pure-glue particle called the scalar glueball. The authors derive a gauge-invariant effective field theory (EFT) for the scalar glueball starting from the nonperturbative Yang–Mills gluon condensate. From that EFT they extract a normalized form factor — a function that describes how the glueball’s interaction strength depends on distance or momentum — and a corresponding interaction radius, a measure of the glueball’s effective size.

To get this EFT the authors model the Yang–Mills vacuum as a coherent condensate of glueball quanta. In that picture the usual gluon field is written as a background part plus a fluctuating part, and the background produces an exponential dressing of the gluon field (an e^{...} factor). That dressing leads to an effective Lagrangian for the scalar glueball and to specific glueball–gluon interaction terms. Using those interactions they build the amputated four-gluon Green’s function (a mathematical object that encodes how four gluons scatter) and then project it into the color-singlet scalar channel with quantum numbers 0^{++} (a spinless, parity-even state).

Projecting onto this scalar channel separates universal kinematic pieces (the parts forced by symmetry and color structure) from the intrinsic glueball dynamics. The remaining intrinsic function is the scalar glueball form factor. From that form factor the authors derive a parameter-free formula for the effective interaction radius of the ground-state scalar glueball, r_int = sqrt(6)/m_φ, which they evaluate using the resonance candidate f_0(1710) to obtain r_int = 0.28 fm. This value is very close to a recent lattice Yang–Mills determination of the glueball mass-radius, quoted as r = 0.263(31) fm. The paper also finds that the predicted momentum dependence of the normalized form factor is consistent with lattice gravitational form factor data. Adding the first excited 0^{++} state changes the interaction radius only modestly (to r_int* = 0.240 fm) and yields stable modifications of the form factor.