SEMICLASSICAL MATRIX-ELEMENTS FOR THE QUARTIC OSCILLATOR

The dilute-instanton-gas approximation leads to unsuppressed (B + L)-violating processes in the standard model at high-energy E and multiplicities N ≥ 1/αw, but is unreliable because of blatant violations of unitarity. A similar and related phenomenon is the fast, unitarity-violating growth of high-multiplicity tree-level fixed-angle cross sections which, typically, for a λφ4 theory behave like E-2λNN!. To explore how unitary is restored by loop corrections we study the large-N limit of amplitudes like <0| x |N> for the d = 1 theory, i.e., for the quartic oscillator. We show that at large N the semiclassical limit is appropriate. We extend the original technique of Landau and find (i) that in cases with two real turning points (single well) the N!λN behaviour is replaced by e-πN for N > O(1/λ) and (i) that in cases with four real turning points (double well), even at energies near the top of the barrier, there is still an exponential barrier factor which equals the square root of the barrier factor at the ground state level. We also describe an alternative, direct approach to the semiclassical limit which is likely to be useful in field theory. It amounts to solving dressed-loop Schringer-Dyson equations, truncated at a certain number of dressed-loops, and using the non-perturbative quantum corrections thereby induced as input to an effective classical of WKB theory. © 1993 Academic Press, Inc.