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Properties of decaying relativistic particles: "GSI anomaly" like effects

K. Urbanowski

Published Jan 1, 2015• 1 citation
OpenAlex

Abstract

We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.

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We study the survival probability of moving relativistic unstable particles and we found the integral representation of the survival amplitude for such particles moving with constant momentum ~ 6 0. Using this representation of the amplitude we found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t/γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities tend to zero as t → ∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the properly computed decay curves of the moving particles have analogous fluctuating form as the decay curves obtained in the GSI experiment.
Late time properties of moving relativistic particles are studied. Within the proper relativistic treatment of the problem we find decay curves of such particles and we show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival amplitude, occur much earlier than it follows from the classical standard approach boiled down to replace time t by t/γL (where γL is the relativistic Lorentz factor) in the formula for the survival probability. The consequence is that fluctuations of the corresponding decay curves can appear much earlier and much more unstable particles have a chance to survive up to these times or later. It is also shown that fluctuations of the instantaneous energy of the moving unstable particles have a similar form as the fluctuations in the particle rest frame but they are seen by the observer in his rest system much earlier than one could expect replacing t by t/γL in the corresponding expressions for this energy and that the amplitude of these fluctuations can be even larger than it follows from the standard approach. All these effects seem to be important when interpreting some accelerator experiments with high energy unstable particles and the like (possible connections of these effects with GSI anomaly are analyzed) and some results of astrophysical observations.
We study the survival probability of moving relativistic unstable particles with definite momentum p⃗≠0\vec{p} \neq 0p​=0. The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time ttt by t/γt/γt/γ (where γγγ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as t→∞t\rightarrow \inftyt→∞ much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime.
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
Results of theoretical studies and numerical calculations presented in the literature suggest that the survival probability P0(t) has the exponential form starting from times much smaller than the lifetime τ up to times t ⪢τ and that P0(t) exhibits inverse power–law behavior at the late time region for times longer than the so–called crossover time T ⪢ τ (The crossover time T is the time when the late time deviations of P0(t) from the exponential form begin to dominate). More detailed analysis of the problem shows that in fact the survival probability P0(t) can not take the pure exponential form at any time interval including times smaller than the lifetime τ or of the order of τ and it has has an oscillating form. We also study the survival probability of moving relativistic unstable particles with definite momentum . These studies show that late time deviations of the survival probability of these particles from the exponential–like form of the decay law, that is the transition times region between exponential–like and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard considerations.
The relativistic quantum decay laws of moving unstable particles are analyzed for a general class of mass distribution densities which behave as power laws near the (nonvanishing) lower bound μ0 of the mass spectrum. The survival probability Pp(t) , the instantaneous mass Mp(t) , and the instantaneous decay rate Γp(t) of the moving unstable particle are evaluated over short and long times for an arbitrary value p of the (constant) linear momentum. The ultrarelativistic and nonrelativistic limits are studied. Over long times, the survival probability Pp(t) is approximately related to the survival probability at rest P0(t) by a scaling law. The scaling law can be interpreted as the effect of the relativistic time dilation if the asymptotic value Mp∞ of the instantaneous mass is considered as the effective mass of the unstable particle over long times. The effective mass has magnitude μ0 at rest and moves with linear momentum p or, equivalently, with constant velocity 1/1+μ02/p2 . The instantaneous decay rate Γp(t) is approximately independent of the linear momentum p , over long times, and, consequently, is approximately invariant by changing reference frame.
The deviation of the decay law from the exponential is a well known effect of quantum mechanics . Here we analyze the relativistic survival probabilities , S ( t , p ) , where p is the momentum of the decaying particle and provide analytical expressions for S ( t , p ) in the exponential (E) as well as the nonexponential (NE) regions at small and large times. Under minimal assumptions on the spectral density function, analytical expressions for the critical times of transition from the NE to the E at small times and the E to NE at large times are derived. The dependence of the decay law on the relativistic Lorentz factor, γ = 1 / 1 − v 2 / c 2 , reveals several interesting features. In the short time regime of the decay law, the critical time, τ s t , shows a steady increase with γ , thus implying a larger NE region for particles decaying in flight. Comparing S ( t , p ) with the well known time dilation formula, e − Γ t / γ , in the exponential region, an expression for the critical γ where S ( t , p ) deviates most from e − Γ t / γ is presented. This is a purely quantum correction. Under particular conditions on the resonance parameters, there also exists a critical γ at large times which decides if the NE region shifts backward or forward in time as compared to that for a particle at rest. All the above analytical results are supported by calculations involving realistic decays of hadrons and leptons .