Scattering in an environment

The cross section of elastic electron-proton scattering taking place in an electron gas is calculated within the closed time path method by means of the resummation of the one-loop self-energy in the photon propagator. Backreaction of the colliding particles on the gas has been taken into account and is found to be dominating the cross section when the energy exchange falls in the vicinity of the Fermi energy. Backreaction reflects the colliding particles-gas entanglement and makes the colliding particle state mixed. The softness of the asymptotic particle-hole states of the gas makes the colliding particle trajectories more consistent and the collision irreversible, rendering in this manner the scattering more classical in this regime.

Figure 1

The leading-order graphs contributing to the transition probability: (a) direct, (b) entanglement contributions. The circle represents the initial density matrix. The CTP-diagonal block of the photon propagator connects charged particle lines on the same time axis in graph (a); the off-diagonal blocks couple the two time axes and their photon lines traverse the final time, the left and right end of graph (b), with periodic boundary conditions in time, i.e., lines ending at the same altitude at the left and right end of the graph are joined.

Figure 2

Two higher-order graphs contributing to the transition probability: (a) direct, (b) entanglement contributions. The particle-hole pairs dress the propagation of the photon in each time axis separately in the direct graph and traverse the final time in the entangled graph.

Figure 3

Logarithm of the differential cross sections (a) $\log {\sigma }_v$, (b) $\log {\sigma }_d$, and (c) $\log ({\sigma }_d+{\sigma }_e)$ as functions of $P/k$ and $\Theta /\pi$.

Figure 4

Spectral strengths (a) $2\pi {\rho }_{\ell }^{(j)}$ of the electric current, (b) $2\pi {\rho }_{\ell }^{(A)}$, and (c) $2\pi {\rho }_t^{(A)}$ of the photon propagator as functions of $|\mathit{q}|/k$ and $z=m{q}^0/k^2$. The additional curve belongs to $\Theta =0.5\pi$.

Figure 5

(a) $D_{\ell }^n$ and contourplots of (b) $D_{\ell }^n$, (c) $2\pi {\rho }_{\ell }^{(A)}$ as functions of $|\mathit{q}|/k$ and $z=m{q}^0/k^2$. The line $\Theta =0.5\pi$ is shown, as well.

Figure 6

(a) $|{\mathcal{T}}_d{|}^2$, (b) $2\pi {\rho }_{\ell }^{(A)}$, (c) ${\Pi }_{\ell }^n$, (d) $2\pi {\rho }_{\ell }^{(j)}$, (e) $|{\mathcal{T}}_d{|}^2+|{\mathcal{T}}_e{|}^2$ (higher curves), $|{\mathcal{T}}_d{|}^2$ (lower curves), and (f) $|{\mathcal{T}}_e{|}^2$ as functions of $P$ at $\Theta =0.3\pi$ (left curve) and $\theta =0.5\pi$ (right curve).

Figure 7

$({\sigma }_d+{\sigma }_e)/{\sigma }_d$ as function of $P/k$ and $\Theta /\pi$.

Figure 8

(a) $D_{\ell }^i$, (b) $D_t^i$ as functions of $P/k$ and $\Theta /\pi$.