A generalization of the Bjorken limit (for the two-point function) to the three-point and four-point functions is given. Some general features of the asymptotic behavior of the $n$-point function are also discussed. These results show that in calculating the various Ward identities for the $n$-point function all currents are "asymptotically conserved." We derive generalized Weinberg sum rules for the three-point functions (these results can be generalized to the $n$-point functions). We show that the $K_L^{}{}_{}{}^0-K_S^{}{}_{}{}^0$ mass difference (in the universal Fermi theory) is quadratically divergent. Making a saturation assumption, we calculate the coefficient of the quadratic divergency and we get a weak-interaction cutoff $\Lambda =4\mathrm{}$ BeV, suggesting that weak interactions are strongly nonlocal. By means of a simple power-counting argument, we find that the $n\mathrm{th}$ order probably behaves like $n!G{\left(G{\Lambda }^2\right)}^{n-1\mathrm{}}$, and assuming that this is some kind of asymptotic expansion, we find that the series begins to blow up for $n\sim {10}^4$. The arguments for this do not constitute a proof. We then study the radiative corrections to the decays $\pi \to e\nu$ and $\pi \to \mu \nu$, which involve a three-point function. We find that these decays cannot be discussed within the framework of current algebra. Finally we show that a somewhat generalized version of the Tamm-Dancoff approximation can be justified if we use our results for the $n$-point function.

• Received 8 March 1968

DOI:https://doi.org/10.1103/PhysRev.175.2165