Pion gravitational form factors in a relativistic theory of composite particles

We extend our relativistic theory of electroweak properties of composite systems to describe simultaneously the gravitational form factors of hadrons. The approach is based on a version of the instant-form relativistic quantum mechanics and makes use of the modified impulse approximation. We exploit the general method of the relativistic invariant parametrizaton of local operators to write the energy-momentum tensor of a particle with an arbitrary spin. We use the obtained results to calculate the gravitational form factors of the pion assuming pointlike constituent quarks. All but one parameters of our first-principle model were fixed previously in works on electromagnetic form factors. The only free parameter, $D_q$, is a characteristic of the gravitational form factor of a constituent quark. The derived form factors of the pion satisfy the constraints given by the general principles of the quantum field theory of hadron structure. The calculated gravitational form factors and gravitational mean-square radius are in a reasonable agreement with known results.

Figure 1

Gravitational $A$ form factor of pion. Full line (red)—the full result; dashed line (blue)—the contribution of the $A$ form factors $g_{10}^{(q)}$ of the constituent quarks; short-dashed line (magenta)—the contribution of the quark $J$ form factors $g_{40}^{(q)}$ (relativistic spin rotation effect).

Figure 2

Our $A$ form factor of pion [full line (red)] in comparison with the result of the authors of [12]. Their $A$ form factor of pion is normalized to its value at zero $t$ (double-dot-dashed, green) line.

Figure 3

The joint of two functions for the pion $D$ form factor for the middle value from (59) and the minimal value from (63). The full line (red)—the $D$ form factor (62); the dashed line (blue)—the solution of (18), (37), (53); the dot-dashed line (black)—$D^{(\pi a)}(t)$; $(-t_c)=2.53{\mathrm{GeV}}^2$.

Figure 4

The dependence of the joint $D$ form factor of pion procedure on the values of the parameter $D_q$ (59) and on the values from the interval (63). The full line (red)—$D_q=-0.1435$, long-dashed line (blue)—$D_q=-0.148$, short-dashed line (magenta)—$D_q=-0.139$. The upper set of curves at $(-t)\sim 1.5{\mathrm{GeV}}^2$—for the minimal value from (63), the lower set—for the maximal value from (63).

Figure 5

The pion gravitational $D$ form factor calculated for the parameters (54), (56) and $D_q=-0.1435$ (59). The full line (red)—the total values obtained using (18), (37), (53). Long-dashed line (blue)—the contribution of quarks $A$ form factors $g_{10}^{(q)}$. Short-dashed line (magenta)—the spin rotation effect (the contribution of the quark $J$ form factor $g_{40}^{(q)}$). Dot-dashed line (black)—the contribution of the quark $D$ form factor $g_{60}^{(q)}$; this curve coincides with $D^{(\pi a)}(t)$.

Figure 6

The pion $D$ form factor calculated for the quark parameters (54), (56), $D_q=-0.1435$ (59) and the minimal value of (63) in comparison with the results of the paper [12]. The full line (red)—our result, double-dot-dashed line (green)—the pion $D$ form factor from [12].