# 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?

edited May 26, 2017

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I haven't done any calculations and I do not know if your equations reflect some physics, but I reasoned in the following way: both fields are "twins" - the same frequency, the same interaction. We may consider them as components of a non relativistic spin 1/2 field, $\omega^2$ being energy $E$. Then the interaction constant $A$ may be considered roughly as a "potential barrier". When the component $\varphi$ collides with it, the component $\kappa$ may appear due to interaction with the barrier. The difference $\omega^2 - A$ may cause propagating waves inside the barrier (real valued wave vector $\bf{k}$), or reflecting waves (evanescent ones inside the barrier)). (I guess the reflecting waves exist in any case, you must admit them.) The physical solutions is chosen by the physically dictated setup - no amplified wave.

answered Feb 2, 2023 by (102 points)
edited Feb 3, 2023

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