Gravitational form factors of heavy quarkonia and their nonrelativistic expansion in a self-consistent light-front quark model

Abstract

We investigate the gravitational form factors of the pseudoscalar heavy quarkonia $η_c(1S,2S)$ and $η_b(1S,2S)$ in a light-front quark model with physical radial wave functions represented by orthogonal mixtures of the two lowest harmonic-oscillator basis states $ϕ_{1S}$ and $ϕ_{2S}$. The Bakamjian--Thomas prescription is applied consistently to constituent matrix elements and Lorentz prefactors at the integrand level. Working in the $Δ^+=0$ frame with $Δ=P'-P$ and $t=Δ^2$, we extract $A_q(t)$ from the quark energy--momentum tensor components $T_q^{++}$, $T_q^{+i}$, and $T_q^{ij}$, and $D_q(t)$ from $T_q^{+-}$, $T_q^{-i}$, and $T_q^{ii}$ using appropriate momentum-transfer projections. The independent extractions agree in the full relativistic calculation and in the analytic nonrelativistic expansions through $ O(v^2)$, where $v=β/m_Q$, $β$ is the Gaussian width, and $m_Q$ is the constituent mass. At zero momentum transfer, the expansions give $D_q^{ϕ_{1S}}(0)=-v^2+ O(v^4)$ and $D_q^{ϕ_{2S}}(0)=-2/3-(13/3)v^2+O(v^4)$. The interference, defined as the sum of the two cross overlaps, first contributes as $D_q^{\mathrm{cross}}(0)=-(4/\sqrt{6})v^2+O(v^4)$. Thus, the pure Gaussian $1S$ contribution is suppressed, whereas a nonzero $2S$ admixture at fixed mixing angle leaves an $O(v^0)$ D-term in the physical ground state. Full numerical results for the mixed states illustrate the effects of heavy-quark flavor and radial structure on mass distributions and internal stresses.