= Mass-moment cancellation of vacuum energy divergences
{title2=$\operatorname{Str}1=\operatorname{Str}m^2=\operatorname{Str}m^4=0$}
For finitely many free physical degrees of freedom with nonnegative <masses>, assign positive weights to <bosons> and negative weights to <fermions>. With a common spherical <ultraviolet cutoff>, the unsubtracted <vacuum energy density> is finite precisely when the three displayed <supertraces> vanish. Expanding $\sqrt{p^2+m^2}=p+m^2/(2p)-m^4/(8p^3)+O(p^{-5})$ proves that they cancel the quartic, quadratic and logarithmic <ultraviolet divergences>. The remaining radial integral converges as $\int^\infty p^{-3}dp$. Equal-mass <boson>–<fermion> pairing is sufficient but stronger than the three conditions; a renormalized <vacuum energy> with a free <counterterm> need not satisfy them.
Back to article page
