Central depression in the charge distributions ofHe3andHe4

Recently Ballot and Fabre de la Ripelle pointed out that the central depression in the "empirical" point proton distribution of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ could be attributed to the two-pion-exchange three-body force. We examine this mechanism by constructing simple phenomenological wave functions of the form of $f\left(x\right)f\left(y\right)f\left(z\right)\mathrm{}{[1-\Sigma g\left(x\right)g\left(y\right)(3\mathrm{}{cos}^2{\theta }_z-1)]}^n$, where $x$, $y$, and $z$ are the interparticle distances, $cos{\theta }_z=\frac{\overrightarrow{\mathrm{x}}·\overrightarrow{\mathrm{y}}}{\mathrm{xy}}$, $n=\frac{1}{2} or 1$, and $\Sigma$ implies the cyclic summation over $x$, $y$, and $z$. The angle dependent term simulates the effect of the two-pion-exchange three-body force. We find that the density near the origin is indeed very sensitive to the angle dependent term of the model wave function, and the central depression can be easily fitted. The analysis is extended to ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, with similar results. We then try to relate the model wave function of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ to the two-pion-exchange three-body force by a variational calculation. About one third of the phenomenological angle dependent term can be obtained, but the result is quite sensitive to the short-range part of the three-body force.