Particle escape into extra space

We focus on escape of a spin integer particle the challenge for which is of course that the corresponding field equation contains the second order time derivative and, in general, may be problematic for interpreting the extra-dimensional part of the field as a wave function for the KK modes as it is usually regarded.

Extra-dimensional models provide a fertile approach to a broad class of problems in the theory of elementary particles and cosmology. In the brane-wirld model building the question of paramount importance is the localization of the Standard Model particles on the brane. We continue our attempt to put forward the theoretical framework for describing the dynamics of particle escape into extra space. The troublesome aspect of our approach presented in [1] has to do with the time evolution of extra-dimensional part of the scalar field. Namely, it does not preserve the norm of the metastable state and therefore contradicts the wave function interpretation. Let us strive to maintain this interpretation of exra-dimensional part of the field as underpinning for the brane-world model and try to describe the time evolution of the quasy-localized state.
To be more concrete in what follows let us focus on the brane-world model with non-factorizable warped geometry [3] where the parameter k is determined by the bulk cosmological and five-dimensional gravitational constants respectively. In making the transition from the bulk with coordinates (x µ , z) to the brane with local coordinates x µ , the field φ(x µ , z) is usually decomposed into four end extra-dimensional parts where ψ m (z) is understood as a wave function responsible for localization of the KK modes [2] with the mass The brane Eq.(1) localizes the massless scalar field as it was shown in [2], but when this field is given bulk mass term the bound state becomes metastable against tunneling into an extra dimension [4]. Let us presume that the metastable states are defined by the Gamow's method as it was done in [4]. namely, if the transverse equation admits tunneling of initially brane localized particle into extra space one finds the complex eigenvalues in the mass spectrum by imposing the outgoing wave boundary condition. To gain a complete picture it should be noticed that for the Gamow's method the concrete form of the time evolution of the state encoded in the Schrödinger equation is a crucial moment. So following the Gamow's method it seems reasonable to take the following ansätze as a satisfactory point of departure where p is the three momentum of the particle, E 0 = m 2 0 + p 2 is its energy and m 0 is defined through the Gamov's method (for simplicity only one metastable mode m 0 is assumed). The four-dimensional plane wave in Eq.(3) stands for a free particle with the mass (p µ p µ ) 1/2 moving in x µ space along p while g(t, z) describes its localization properties across the brane which, in general, may vary in course of time. For this ansätze one gets the following equation where µ is the bulk mass parameter [4]. Following the paper [4] under assumption µ ≪ k one finds without much ado and the corresponding wave function satisfying the outgoing wave boundary condition. In this way one finds that the probability of finding the particle on the brane decays exponentially in time and the corresponding lifetime is given by ∼ p 2 + m 2 0 /m 0 Γ. But, as it is well known the precise quantum mechanical consideration of metastable state decay based on the approach proposed by Fock and Krylov [5] leads to the deviation from exponential decay law for small and large values of time. More precisely in quantum mechanics it is well established that an exponential decay cannot last forever if the Hamiltonian is bounded below and cannot occur for small times if, besides that, the energy expectation value of the initial state is finite [6]. Let us consider this problem for the brane-world model following to the general ansätze φ = e i(E0t− p x) g(t, z) .

The equation of motion for
f (t, z) = e iE0t g(t, z) takes the form the general solution for which is given by [7] where satisfying the junction condition across the brane with the index ν = 4 + µ 2 /k 2 and coefficients , To compute f (t, z) one needs to know the initial data f (0, z),ḟ (0, z). In general, arbitrarily taking the initial velocityḟ (0, z) we face the problem that the norm of f (t, z) may not be preserved [1].
With the above comment in mind, to keep as close as possible to the time dependence of Schrödinger equation, the further insight into the evaluation of escape dynamics can be gained by considering the following solution where g(0, z) denotes the brane localized initial state. So that the resulting prescription for evaluating the probability of particle to be confined at instant t on the brane is similar to the quantum mechanical one considered in [5,6]. Correspondingly the time evolution of the decay can be divided into three domains as in the quantum mechanical case indicated above. Under assumption µ/k ≪ 1 one finds that the function |C(E)| 2 where has a simple pole in the fourth quadrant of a complex E plane [1,4] Since, when t > 0 and |E| → ∞, e −iEt → 0 in the fourth quadrant, for evaluating of the transition amplitude it is convenient to deform the integration contour as it is shown in Fig.1. In this way one finds where the residue term determines the exponential decay law with the decay width ∼ Γm 0 / p 2 + m 2 0 . In the vicinity of the polem = m 0 − iΓ Thus the pre-exponential term that comes from the residue at the pole does not depend on the three momentum p. By taking into account that µ/k ≪ 1 and the localization width of the scalar is ∼ k −1 , one can simply take So using the solution Eq.(8), on the quite general grounds as in the standard quantum mechanical case [6] one concludes that initially the decay is slower than exponential, then comes the exponential region and after a long time it obeys a power law. We think the Eq.(8) is of particular importance for describing the dynamics of particle escape into extra space. In studying this problem for arbitrary spin integer particle one can construct analogous solution.
Let us make a brief clarification of the above discussion. Usually in constructing the relativistic quantum mechanics the square root from the Hamiltonian of a relativistic free particle is removed. The resulting Klein-Gordon equation is altered from the Schrödinger equation in that it contains the second order time derivative that in general proves impossible the probability interpretation of the solution. Knowing the eigenfunction spectrum loosely speaking one can construct the solution of Eq.(11) ψ(t) = dEe −iEt ϕ(E) ϕ(E)|ψ(0) , satisfying the Eq.(12) as well. This solution admits a straightforward probability interpretation as in the Schrödinger case and is thereby unique. This is essentially what we have done in the present paper for describing the time evolution of the quasilocalized scalar particle on the brane. This brief paper corrects and complements our previous consideration [1].