Hisashi Kikuchi

Valley views: instantons, large order behaviors, and supersymmetry

Abstract The elucidation of the properties of the instantons in the topologically trivial sector has been a long-standing puzzle. Here we claim that the properties can be summarized in terms of the geometrical structure in the configuration space, the valley. Evidence for this claim is presented in various ways. The conventional perturbation theory and the non-perturbatioe calculation are unified, and the ambiguity of the Borel transform of the perturbation series is removed. A ‘proof’ of Bogomolnys “trick” is presented, which enables us to go beyond the dilute-gas approximation. The prediction of the large order behavior of perturbation theory is confirmed by explicit calculations, in some cases to the 478th order. A new type of supersymmetry is found as a by-product, and our result is shown to be consistent with the non-renormalization theorem. The prediction of the energy levels is confirmed with numerical solutions of the Schrodinger equation.

Recent Developments of the Theory of Tunneling

Path-integral approach in imaginary and complex time has been proven successful in treating the tunneling phenomena in quantum mechanics and quantum field theories. Latest developments in this field, the proper valley method in imaginary time, its application to various quantum systems, complex time formalism, asympton theory for the large order analysis of the perturbation theory, are reviewed in a self-contained manner.

Unitarity restoration in the multi-instanton process

Abstract The instanton induced multi-particle scattering amplitude is considered in the scalar sector of standard electroweak theory. The Feynman diagram expansion is obtained, where the internal lines come from the interaction between instantons and/or anti-instantons. The resulting amplitude is manifestly unitary. An example of the unitary amplitude is obtained using the K -matrix formalism.

Valleys in quantum mechanics

Abstract Conventionally, perturbative and non-perturbative calcula-tions are performed independently. In this paper, valleys in the configuration space in quantum mechanics are investigated as a way to treat them in a unified manner. All the known results of the interplay of them are reproduced naturally. The prescription for separating the non-perturbative contribution from the perturbative is given in terms of the analytic continuation of the valley parameter. Our method is illustrated on a new series of examples with the asymmetric double-well potential. We obtain the non-perturbative part explicitly, which leads to the prediction of the large order behavior of the perturbative series. We calculate the first 200 perturbative coefficients for a wide range of parameters and confirm the agreement with the prediction of the valley method.

FAKE INSTABILITY IN THE EUCLIDEAN FORMALISM OF QUANTUM TUNNELING

The Euclidean path-integral formalism is studied for the system without any unstable state. While the usual bounce calculation yields a fake complex energy spectrum, proper account of the global structure of the functional space is shown to lead to the real eigenvalues. For this purpose, the valley instanton is constructed by the proper valley method and the dilute-gas approximation is carried out. This establishes a general imaginary-time formalism, unifying the instanton and the bounce method. {copyright} {ital 1997} {ital The American Physical Society}

Decay of Z into two light Higgs bosons.

If the standard electroweak gauge model is extended to include two or more Higgs doublets, there may be a neutral Higgs boson {ital h} which is light (with a mass of say 10 GeV) but the {ital hZZ} coupling is suppressed so that it has so far escaped experimental detection. However, the effective {ital hhZZ} coupling is generally unsuppressed; hence, the decay of {ital Z} into two light Higgs bosons plus a fermion-antifermion pair may have an observable branching fraction, especially if {ital h} decays invisibly as, for example, in the recently proposed doublet Majoron model.

INTERACTING INSTANTONS FOR TeV PHYSICS

Multi-instanton effect in the standard electroweak model for baryon and lepton number violation is discussed. Instanton interaction is shown to play a major role in the TeV range. First, the validity of the treatment of the interaction is explicitly checked in the quantum-mechanical case. Then interactions induced by both bosons and fermons are examined in the SU(2) Yang–Mills–Higgs model. They are shown to lead to an effective theory which is manifestly unitary. This results in an estimate that the relevant cross section saturates the unitary bound at the TeV scale, indicating the possibility of detecting baryon- and lepton-number-violating phenomena.

The fermion determinant, its modulus and phase

We consider path integration of a fermionic oscillator with a one-parameter family of boundary conditions with respect to the time coordinate. The dependence of the fermion determinant on these boundary conditions is derived in a closed form with the help of the self-adjoint extension of differential operators. The result reveals its crucial dependence on them, contrary to the conventional understanding that this dependence becomes negligible over sufficiently long time evolution. An example in which such dependence plays a significant role is discussed in a model of supersymmetric quantum mechanics.

Enhancement of the Transition Magnetic Moments of a Neutrino by Degenerate Electrons

The one-loop induced magnetic dipole moments of a neutrino are examined in a background of degenerate electrons in the standard model. For the nonrelativistic neutrino, they are enhanced by a factor \( (8\pF/3\mnu) \), where \pF\ is the electron Fermi momentum and \mnu\ the neutrino mass. For the relativistic neutrino, they enhance the flavor-changing but helicity-conserving process because of the absence of the GIM cancellation.