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We study the quantum field theory of a charged $\phi^4$ field in latticeregularization at finite density and low temperature in 2 and 4 dimensions withthe goal of analyzing the connection of condensation phenomena to scatteringdata in a non-perturbative way. The sign problem of the theory at non-zerochemical potential $\mu$ is overcome by using a worldline representation forthe Monte Carlo simulation. At low temperature we study the particle number asa function of $\mu$ and observe the steps for $\mbox{1-,}$ 2- and 3-particlecondensation. We determine the corresponding critical values $\mu_n^{crit}, \,n = 1,2,3$ and analyze their dependence on the spatial extent $L$ of thelattice. Linear combinations of the $\mu_n^{crit}$ give the interactionenergies in the 2- and 3-particle sectors and their dependence on $L$ isrelated to scattering data by L\"uscher's formula and its generalizations tothree particles. For 2-$d$ we determine the scattering phase shift and for4-$d$ the scattering length. We cross-check our results with a determination ofthe mass and the 2- and 3-particle energies from conventional 2-, 4-, and6-point correlators at zero chemical potential. The letter demonstrates thatthe physics of condensation at finite density and low temperature is closelyrelated to scattering data of a quantum field theory.

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