3-LOOP ON-SHELL CHARGE RENORMALIZATION WITHOUT INTEGRATION - LAMBDA-QED(MS)BAR TO 4 LOOPS

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavour D-dimensional QED, in terms of GAMMA-functions and a single F32 series, whose expansion near D = 4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limit D --> 4, we find that the MSBAR coupling alphaBAR(mu) satisfies the boundary condition [GRAPHICS] where m is the physical lepton mass and alpha is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the MS beta-function, we obtain GRAPHICS] at the four-loop level of one-flavour QED.