Induced gravity and Planck zeros

Starting with an asymptotically free gauge theory with dynamical symmetry breaking and a mass hierarchy, we investigate the Adler-Zee formula for the induced gravitational constant. We study the two-point function psi(q2), constructed with the trace of the energy-momentum tensor. First, we show that if the zeros of psi are at a mass scale significantly below the leading scale, then G/sub ind/ -1 = 0 (m/sub zero/ 2) making it impossible to get a realistic G/sub ind/ from the Adler-Zee formula with low-mass zeros. Next we use the Jensen formula to derive a sum rule for Vertical Barm/sub zero/Vertical Bar. The analysis of this sum rule coupled with the result above leads to a dilemma with only one reasonable resolution. To get a realistic G/sub ind/ from the Adler-Zee formula, psi(q2) must have a pair of complex-conjugate zeros at q2 = M02 +- 2iνM0, where M0 is large and of the maximal scale and ν/M0<<1. The presence of this zero essentially determines G/sub ind/ -1. It gives a lower bound, which with our previously derived general upper bound gives [π2/4(ln10)288] C/sub psi/M02< or =(16πG)-1 < or = (5π2/288) C/sub psi/M02, where C/sub psi/ is the anomaly coefficient, a number easily determined by low-order perturbation theory for any group