The gluonic condensate from the hyperfine splitting $M_{\mathrm{cog}}({\chi }_{cJ})-M(h_c)$ in charmonium

Abstract

The precision measurement of the hyperfine splitting ${\Delta }_{\mathrm{HF}}(1P,c\bar{c})=M_{\mathrm{cog}}({\chi }_{cJ})-M(h_c)=-0.5\pm 0.4$ MeV in the Fermilab--E835 experiment allows to determine the gluonic condensate $G_2$ with high accuracy if the gluonic correlation length $T_g$ is fixed. In our calculations the negative value of ${\Delta }_{\mathrm{HF}}=-0.3\pm 0.4$ MeV is obtained only if the relatively small $T_g=0.16$ fm and $G_2=0.065(3)$ GeV${}^4$ are taken. These values correspond to the ``physical'' string tension $(\sigma \approx 0.18$ GeV${}^2$). For $T_g\ge 0.2$ fm the hyperfine splitting is positive and grows for increasing $T_g$. In particular for $T_g=0.2$ fm and $G_2=0.041(2)$ GeV${}^4$ the splitting ${\Delta }_{\mathrm{HF}}=1.4(2)$ MeV is obtained, which is in accord with the recent CLEO result.

Comments: 9 pages revtex 4, no figures