The Contribution of Electron-Positron Pair Production to the Vacuum   Energy
  The vacuum, defined as the state where no particles can be observed, is interpreted here to imply that the lifetime of the e-p pair should be equal to the Planck time. Concerning the title's subject, a perfect theory would require that the true vacuum expectation value of the operator associated with pair production, be compatible with the normalization of the true vacuum. At present, a calculation of the vacuum energy based on Feynman diagrams reveals a serious difficulty: if only second order terms of the S-matrix are retained, and because there are no external lines, it follows that the space-time integrations over the coordinates, involved in the calculation of the vacuum expectation value of the S-matrix, give rise to two identical delta functions: the amplitude is thus proportional to the space-time volume of integration, L4. The square of the amplitude defies then any physically meaningful interpretation. One is faced here with two evils: modify the interaction Lagrangian so that the amplitude becomes proportional to L2, or abstain from any calculation. It is felt that the first one is the lesser evil. If the square of the amplitude is proportional to L4 (instead of L8), it, can be interpreted as being the number, N, of events (pairs created), in the volume of integration. In the calculations for N it was assumed that the integral over momentums (rescaled to be dimensionless) was of the order unity, and that processes with small virtual photon-energy are predominant. The pairs' contribution to the vacuum density is then given by the mass of the particles multiplied, by the number of events per unit volume and unit time, as well as, by the pairs lifetime. It is found that the calculated value for the vacuum density is in surprisingly good agreement with the observations. 
Bernard R. Durney
physics.gen-ph