On the Relation Between Time Representations and Inner Product Spaces

Stable states (particles), ghosts and unstable states (particles) come with different types of time representations in unitary groups—definite or indefinite. These representations are discussed with respect to the induced inner product spaces as extensions of Hilbert spaces. Unstable particles with their decay channels are treated as higher dimensional probability collectives.     

An analysis of mean life and lifetime of unstable elementary particles

A theoretical analysis of the concept of lifetime and mean life of unstable elementary particles is presented. New analytic formulas for lifetime and mean life as a function of decay width and the mass of unstable particle are derived for Breit-Wigner and Matthews-Salam energy distributions. It is demonstrated that, for unstable particles with a larger width or decay energy threshold, the deviation from the generally accepted mean life ^m = ^–1 is significant. The behavior of the decay law P(t) for small times is analyzed, and it is shown that the Breit-Wigner distribution violates the condition P(t = 0) = 0, whereas the Matthews-Salam distribution satisfies it.     

The Markovian master equations for unstable particles

The theory of Markovian master equations is applied to a certain model of unstable particles. The exponential decay law is obtained in the weak coupling limit. The connection to the method of one-parameter contracting semigroups on a single particle Hilbert space is given.     

Non-exponential Decay in Quantum Field Theory and in Quantum Mechanics: The Case of Two (or More) Decay Channels

We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is exponential) and equals the quantity Γ ₁/Γ ₂, where Γ ₁ and Γ ₂ are the respective tree-level decay widths. However, in the full treatment (both for QFT and QM) it is an oscillating function around the mean value Γ ₁/Γ ₂ and the deviations from this mean value can be sizable. Technically, we study the decay properties in QFT in the context of a superrenormalizable Lagrangian with scalar particles and in QM in the context of Lee Hamiltonians, which deliver formally analogous expressions to the QFT case.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Integrable scattering theories with unstable particles

We formulate a new bootstrap principle which allows for the construction of particle spectra involving unstable as well as stable particles. We comment on the general Lie algebraic structure which underlies theories with unstable particles and propose several new scattering matrices. We find a new Lie algebraic decoupling rule, which predicts the renormalization group flow in dependence of the relative ordering of the resonance parameters. The proposals are exemplified for some concrete theories which involve unstable particles, such as the homogeneous sine-Gordon models and their generalizations. The new decoupling rule can be validated by means of our new bootstrap principle and also via the renormalization group flow, which we obtain from a thermodynamic Bethe ansatz analysis.O.A. Castro-Alvaredo: Present addressLaboratoire de Physique, Ecole Normale Supérieure de Lyon, Allée dItalie, 69364 Lyon Cedex, FranceJ. Dreißig: Present address Universität Potsdam, Institut für Physik, Am Neuen Palais 10, 14469 Potsdam, GermanyA. Fring: Present address Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB, UK     

Decays of Z Bosons into Triple-Quark and Double-Quark Structures

This paper presents a mechanism of decay of elementary particles, which has been developed based on a new approach. The results obtained may be of interest to experimenters engaged in the decay of Z ⁰ bosons. Schemes and modes of the structural decay of gamma structures, one of which is the Z ⁰ boson, are given. Analysis of the schemes and modes of the structural decay has shown that the decays of the product triple-quark particles (products of the decay of a Z ⁰ boson into triple-quark particles) result in the occurrence of a new mode of decay – the decay of one triple-quark particle into three triple-quark particles. According to the proposed mechanism of the decay of particles, an electron, a muon, a triton, and a gamma photon are particles consisting of a quark–antiquark pair and having a spin equal to unity.     

An optimised molecular model for ammonia

Based on molecular dynamics (MD) computer simulations we investigate the dynamic behaviour of a model complex fluid suspension consisting of large (A) particles (the ‘solute’) immersed in a bath of smaller ‘solvent’ (B) particles. The goal is to identify the effect of systematic simplifications (coarse-graining) of the solvent on typical microscopic time correlation functions characterizing the single-particle and collective dynamics of the solute. As a reference system we employ a binary Lennard–Jones mixture of spherical particles with significant differences in particle sizes (σ_A>σ_B) and masses (m _A>m _B). We then replace the original B particles step by step by a reduced number of larger and heavier particles such that the mass and volume fraction of B particles is kept constant. At each step of coarse-graining, the intermolecular interactions between A particles are chosen such that the static A–A structure of the reference system is preserved. Our MD results indicate that coarse-graining has a profound influence on both the single-particle dynamics as reflected by the self-diffusion constant and the collective dynamics represented by the distinct part of the van Hove time correlation function. The latter holds only at intermediate packing fractions, whereas the collective dynamics turns out to be essentially insensitive to coarse-graining at high packing fractions.     

Decay Law of Moving Unstable Particle

Quantum relativistic decay law of moving unstable particle is analytically calculated in the model case of the Breit–Wigner mass distribution. It turns out that Einstein time dilation of the moving particle decay holds approximately at times when the decay is exponential. The related correction is calculated analytically. Being very small at these times it is practically unobservable. It is shown that Einstein dilation fails for large times t when decay is not exponential. An unstable system of the kind of K ₀-meson (which is the superposition of K _s and K _I) is also considered. In this case, the violation of Einstein dilation is shown to be appreciable at all times under some condition     

Two interacting particles in a disordered chain I: Multifractality of the interaction matrix elements

For N interacting particles in a one dimensional random potential, we study the structure of the corresponding network in Hilbert space. The states without interaction play the role of the “sites”. The hopping terms are induced by the interaction. When the one body states are localized, we numerically find that the set of directly connected “sites” is multifractal. For the case of two interacting particles, the fractal dimension associated to the second moment of the hopping term is shown to characterize the Golden rule decay of the non interacting states and the enhancement factor of the localization length. Received: 17 April 1998 / Accepted: 14 May 1998     

Particle spectrum and geometrical symmetry

The spectrum of unstable particles is studied on a model meeting the requirement of geometrical symmetry expressed by the restricted Lorentz groupL _r, which is represented by an unstable model particle described by the invariant tensor or spintensor of the groupL _r satisfying the Klein-Gordon equation. The problem of the spectrum of model particles is formulated and treated as a certain eigenvalue problem invariant with regard toL _r. The calculated spectrum of the reduced levels mass/width of the model particles is spin independent, agrees with the observed spectrum of resonances and shows that the model employed represents certain laws manifesting themselves in the observed spectrum of unstable particles.In conclusion the author would like to thank Dr. J. Fischer for fruitful discussions on this work and Dr. K. Kunc for performing some numerical computations on the computer.     

Time reversal for unstable particles

The time reversal operator (T) for unstable particles is defined such that the fact of decay of the particle is not mistaken for a dynamical violation of T in the decay. This definition is not valid where the medium in which the decay takes place may have its own intrinsic T violation. Based on this limitation, some effort is made to restrict the conclusions drawn from apparent T violation in K meson decay.     

Factorization method in the model of unstable particles with a smeared mass

The method of factorization, based on the model of unstable particles with a smeared mass, is applied to the processes with an unstable particle in the intermediate state. It was shown, that in the framework of the method suggested, the decay rate and cross section can be represented in the universal factorized form for an arbitrary set of particles. An exact factorization is caused by the specific structure of unstable particles propagators. We performed the phenomenological analysis of the factorization effect. The text was submitted by the authors in English.     

On an intermediate structure of giant analog resonances in medium and heavy nuclei

The effect of coupling of the usual particle-hole states to particle-hole states built above a collective state on the decay characteristics and structure of a giant resonance with higher isospin (T _>) has been estimated. Numerical calculations are carried out for⁶⁰Ni,⁹⁰Zr and²⁰⁸Pb.     

Excitation of lowest 0+-states in two-neutron transfer reactions in even actinides

The spectroscopic factor is considered for two-nucleon transfer reactions involving the excitation of quadrupole one-phonon states. The particle number conservation condition for the collective phonon state is chosen in the form of the adiabatic theories with the moving frame. The variation of the energy gap in the collective phonon state is taken into account as well as the difference of the mean numbers of particles in the initial and final nuclei. Numerical calculations of the spectroscopic factor are carried out for the (p, t)- and (t, p)-reactions for a number of actinide nuclei.     

CP violation in correlated production and decay of unstable particles

We study resonant CP-violating Einstein-Podolsky-Rosen correlations that may take place in the production and decay of unstable scalar particles at high-energy colliders. We show that as a consequence of unitarity and CPT invariance of the S-matrix, in 2→2 scatterings mediated by mixed scalar particles, at least three linearly independent decay matrices associated with the unstable scalar states are needed to obtain non-zero CP-odd observables that are also odd under C-conjugation. Instead, for the correlated production and decay of two unstable particle systems in 2→4 processes, we find that only two independent decay matrices are sufficient to induce a net non-vanishing CP-violating phenomenon. As an application of this theorem, we present numerical estimates of CP asymmetries for the correlated production and decay of supersymmetric scalar top-anti-top pairs at the LHC, and demonstrate that these could reach values of order one. As a byproduct of our analysis, we develop a novel spinorial trace technique, which enables us to efficiently evaluate lengthy expressions of squared amplitudes describing the resonant scalar transitions.     

Azimuthal Correlations of Fragments with Light Charged Particles in Reaction of 25MeV/u 40Ar+115In

The large angle correlations for in-plane and out-of-plane have been measured for the pairs of the fragments and light charged particles (LCPs) in the reaction of 25MeV/u ⁴⁰Ar+¹¹⁵In. The azimuthal correlation functions and the azimuthal asymmetry factors were obtained. The azimuthal correlation functions of all pairs between fragments and a particles are of minimum value in φ=90°plane. It indicates that the LCPs and fragments formed in the reaction exhibit an enhanced emission in the reaction plane due to collective rotation-like effect induced by attractive mean field. The more heavier the masses of the coincident LCPs and fragments are, the more stronger the left-right asymmetries of the coincident particles with respect to the beam direction in the reaction plane are, the more preferential the particle emission to the direction opposite to the coincident reaction products is. Along with the increase of the mass of the coincident LCPs and fragments, the influences of the sequential decay and particle final state interactions on the azimuthal correlation functions of the correlated pairs in the φ=0° plane decrease and vanish at last, the collective rotation-like effect is enhanced, and the azimuthal asymmetries increase.     

Meson-baryon femtoscopy in Au+Au collisions at 200 GeV measured by STAR experiment

Correlations between non-identical particles at small relative velocity probe asymmetries in the average space-time emission points at freezeout [1]. Such asymmetries may arise from long-lived resonances, bulk collective effects, or differences in the freezeout scenario for the different particle species. STAR has extracted pion-proton correlation functions from a high-statistics dataset of Au+Au collisions at ?{s_NN }\sqrt {s_{NN} } = 200 GeV. We present a femtoscopic analysis of this data for all combinations of charged pions and (anti-) protons, for collisions of different centrality. The measurements are compared with calculations of a simple Blast-wave model, in which asymmetries are driven only by collective flow, as well as with Therminator [2], which also accounts fully for resonance effects.     

Lamb shift in muonic hydrogen—II. Analysis of the discrepancy of theory and experiment

Currently, both the g factor measurement of the muon as well as the Lamb shift 2S–2P measurement in muonic hydrogen are in disagreement with theory. Here, we investigate possible theoretical explanations, including proton structure effects and small modifications of the vacuum polarization potential. In particular, we investigate a conceivable small modification of the spectral function of vacuum polarization in between the electron and muon energy scales due to a virtual millicharged particle and due to an unstable vector boson originating from a hidden sector of an extended standard model. We find that a virtual millicharged particle which could explain the muonic Lamb shift discrepancy alters theoretical predictions for the muon anomalous magnetic moment by many standard deviations and therefore is in conflict with experiment. Also, we find no parameterizations of an unstable virtual vector boson which could simultaneously explain both “muonic” discrepancies without significantly altering theoretical predictions for electronic hydrogen, where theory and experiment currently are in excellent agreement. A process-dependent correction involving electron screening is evaluated to have the right sign and order-of-magnitude to explain the observed effect in muonic hydrogen. Additional experimental evidence from light muonic atoms and ions is needed in order to reach further clarification.