The gluonic condensate from the hyperfine splitting Mcog(XcJ) — M(hc) in charmonium

The precision measurement of the hyperfine splitting, ΔHF(1P, c-c) = Mcog(xcJ) — M(hc), in the Fermilab-E835 and CLEO experiments allows one to determine the gluonic condensate G2 with high accuracy if the gluonic correlation length Tg is fixed. In our calculations, the negative value of ΔHF = −0.5 ± 0.4 MeV, as in the E835 experiment, is obtained only if the relatively small Tg = 0.16 im and G2 = 0.060(3) GeV4 are taken. For Tg ≥ 0.2 fm, the hyperfine splitting is positive and grows for increasing Tg. In particular, for Tg = 0.2 fm and G2 = 0.045(2) GeV4, the splitting ΔHF = 1.0(5) MeV is just in accordance with the recent CLEO result. The values of G2 taken correspond to the “physical” string tension σ ≈ 0.18 GeV2.