Finite density condensation and scattering data - a study in $\phi^4$ lattice field theory

We study the quantum field theory of a charged $\phi^4$ field in lattice regularization at finite density and low temperature in 2 and 4 dimensions with the goal of analyzing the connection of condensation phenomena to scattering data in a non-perturbative way. The sign problem of the theory at non-zero chemical potential $\mu$ is overcome by using a worldline representation for the Monte Carlo simulation. At low temperature we study the particle number as a function of $\mu$ and observe the steps for $\mbox{1-,}$ 2- and 3-particle condensation. We determine the corresponding critical values $\mu_n^{crit}, \, n = 1,2,3$ and analyze their dependence on the spatial extent $L$ of the lattice. Linear combinations of the $\mu_n^{crit}$ give the interaction energies in the 2- and 3-particle sectors and their dependence on $L$ is related to scattering data by L\"uscher's formula and its generalizations to three particles. For 2-$d$ we determine the scattering phase shift and for 4-$d$ the scattering length. We cross-check our results with a determination of the mass and the 2- and 3-particle energies from conventional 2-, 4-, and 6-point correlators at zero chemical potential. The letter demonstrates that the physics of condensation at finite density and low temperature is closely related to scattering data of a quantum field theory.