Hyperfine splitting for the ground-state muonic3He atom-corrections up to alpha2

The author examines the relativistic, QED and nuclear finite-size corrections of the hyperfine splitting for ground-state muonic 3 He up to alpha 2 . The author finds that the nuclear finite-size correction and the magnetic correction to the electron wavefunction of the hyperfine splitting, Delta v e , which accounts for the magnetic interaction between the electron and the 3 He nucleus, are quite different from the previous results of Huang and Hughes (1979,1980,1982). The author obtains the total hyperfine splitting, Delta v=4166.540 MHz, with the uncertainty, +or-0.001 MHz, which only originates from the calculation of the Schrodinger wavefunction, and the uncertainty, +or-0.004 MHz, which originates from the neglect of order alpha 3 and above in calculating the relativistic, QED and nuclear finite-size correction, and the uncertainties of the parameters of the electric and magnetic form factors of the 3 He nucleus. This result is more consistent with the experimental result, 4166.3+or-0.2 MHz, than the previous theoretical results.