Entanglement between Quarks in Hadrons

Abstract

Entanglement between valence quarks and the sea in hadrons is shown to be deeply connectedto important features of QCD, namely chiral symmetry breaking and confinement. The sea degrees of freedom can be traced over to give a reduced density matrix, and it is shown that the resulting entanglement entropy acts as an order parameter of chiral symmetry breaking and potentially confinement in the nucleon. A general decomposition of the nucleon state in components labeled by definite chiral transformation properties is considered. In particular, the most general expression of the nucleon state in the chiral basis with three valence quarks is constructed, consistent with known QCD constraints, and its properties are studied. In the chiral basis, the nucleon state is naturally a bipartite system where all non-valence spin can be traced over to give a reduced density matrix for the valence spin. It is shown that the resulting entanglement entropy acts as an order parameter of chiral symmetry breaking on the null-plane. In the large-Nc limit, the entanglement entropy is minimized and the valence spin accounts for all of the nucleon spin, while in the limit of maximal entanglement entropy, the nucleon loses all memory of the valence spin and is therefore entirely accounted for by the non-valence spin. This suggests that the small observed valence content of the proton spin is a signature of strong entanglement in the nucleon between the valence quarks and the sea. The nucleon state vector in the chiral basis, fit to low-energy data, gives a valence spin content consistent with experiment and lattice QCD determinations, and has large entanglement entropy. Entanglement between valence and sea quarks in 1+1d QCD is also considered. While predecessors to QCD such as the parton model had a clear distinction between valence and sea particles, the Fock state expansion of a hadron gives no unambiguous definition of which quarks are valence quarks and which belong to the parton sea aside from their flavor. A rigorous definition of valence-sea (VS) entanglement in QCD is developed, which is consistent with the observation that it vanishes in the large-Nc limit and will remain low when finite-Nc states resemble their large-Nc counterparts. We perform a numerical study of VS entanglement in 1+1 dimensional discrete light-cone quantized QCD, and in the process develop a method for building the color-singlet basis of 1+1d QCD that is manifestly complete and orthogonal by construction. We find that the VS entanglement entropy for the first few excited states of both mesons and baryons is relatively low compared to all other states in the spectrum, with the VS entropy of ground state hadrons providing a minimum. We also see that for ground state mesons the entropy is well described in the 1/Nc approximation. These results suggest that low energy hadrons may be the only QCD bound states for which the large-Nc expansion, and perhaps the parton model, provide an accurate description. This work also provides evidence that the VS entanglement entropy of QCD in 3+1d, which would likely serve as an order parameter for the QCD phase transition, may be perturbatively accessible through a large-Nc expansion, and we develop a qualitative picture of what the VS entanglement entropy of nucleons in real QCD may look like.