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Special question about particle-particle oscillations

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Suppose we have two scalar fields $\varphi, \kappa$. Next, suppose there is a region in space where they are mix with each other, i.e., we have a lagrangian
$$
\tag 1
L_{\text{int}} = A \varphi \kappa
$$
By taking into account their kinetic term, we have following EOMS:
$$
\left(\omega^{2} + \partial_{\mathbf{r}}^2 - \begin{pmatrix}0 & A \\ A & 0\end{pmatrix}\right)\begin{pmatrix}\varphi\\ \kappa\end{pmatrix} = 0
$$
It gives rise to particle oscillations. 

Next, suppose we have a beam of $\varphi$ particles propagating along $z$ axis. After entering the domain (say, at $z=0$) in which there is the interaction $(1)$ it begins to oscillate into $\kappa$ particle. I want to calculate the probability of oscillation at $z>0$. It turns out that it is proportional to
$$
P_{\varphi\to\kappa}\sim |e^{-ik_{+}z}-e^{-ik_{-}z}|, \quad k_{\pm} = \sqrt{\omega^2 \mp A} 
$$ 
It turns out that for $|A|>\omega$ one of the momenta $k_{+}$, $k_{-}$ becomes imaginary, and the probability doesn't behave as oscillating function, but instead is exponentially amplified or damped.

What is the physical reason for this?

asked May 26 in Theoretical Physics by NAME_XXX (1,010 points) [ revision history ]
edited May 26 by NAME_XXX

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