Recalculation of the unitary single planar dual loop in the critical dimension of space time

The single planar dual-loop amplitudes are recalculated in the critical dimension of space time paying particular attention to the unitarity property by ensuring that the only states propagating internally in the loop are those needed to factorize tree diagram residues in a positive definite way. The two new technical features which make this possible are (i) a newly discovered physical state projection operator valid if the space- time dimension takes the critical value; (ii) the use of Feynman's tree theorem whereby it is sufficient to have the above projection operator on the mass shell. The final result agrees with a previous conjecture in that it differs from the original calculations with unprojected propagators by two inverse power of a certain partition function. The results apply to both the ordinary dual model and the Neveu-Schwarz model. (12 refs).