Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entanglement and Vacuum Modification

The production of jets should allow testing the real-time response of the QCD vacuum disturbed by the propagation of high-momentum color charges. Addressing this problem theoretically requires a real-time, nonperturbative method. It is well known that the Schwinger model [QED in ($1+1$) dimensions] shares many common properties with QCD, including confinement, chiral symmetry breaking, and the existence of vacuum fermion condensate. As a step in developing such an approach, we report here on fully quantum simulations of a massive Schwinger model coupled to external sources representing quark and antiquark jets as produced in $e^{+}{e}^{-}$ annihilation. We study, for the first time, the modification of the vacuum chiral condensate by the propagating jets and the quantum entanglement between the fragmenting jets. Our results indicate strong entanglement between the fragmentation products of the two jets at rapidity separations $\mathrm{\Delta }\eta \le 2$, which can potentially exist also in QCD and can be studied in experiments.

Figure 1

Left: Time evolution of the local charge density (vertical bars) and of the electric field (arrows), with vacuum expectation values subtracted. Black (white) even (odd) sites correspond to (anti)fermions. The position of the external sources is shown above each configuration. From top to bottom, the rows are for time values (in units of lattice spacing $a$) $t/a=2^{-}$ to ${10}^{-}$, where $n^{-}\equiv n-ϵ$ with $ϵ$ being an arbitrarily small positive number. Right: From top to bottom, time evolution of electric energy, scalar fermion density, entanglement entropy, and electric charge. Dotted lines in the first panel show the electric energy generated by the external sources. The value of the vacuum fermion condensate integrated over the lattice length is ${\nu }_0=-5.16$.

Figure 2

Illustration of correlated and uncorrelated measurements of two point correlation functions. The uncorrelated setup is obtained as an uncorrelated linear superposition of jets created by a single (anti)fermion source moving to the (left) right.

Figure 3

Time evolution of two-point correlation functions with various separations. The upper (lower) panel is for a correlated (uncorrelated) setup. The large difference between the two cases is a signature of quantum entanglement in the produced pairs. Insert: Spatial-rapidity dependence of the two-point correlation at the end of the evolution.