Parton wave functions, density matrices and entropy in DGLAP evolution

Abstract

Parton multiplicity entropy, proposed by Kharzeev and Levin as entanglement between the region probed in deep inelastic scattering and the proton's remainder, was compared with hadron multiplicity data. It is computed from parton densities, which do not fix the partons' quantum state. We show that a one-parton density matrix from Lipatov's wave functions determines two entropies for resolved modes with finite mean number and entropy. Its trace is the mean parton number, and normalization gives a sampled parton's entropy. A compatible thermal state, if it exists, maximizes entropy at fixed one-parton matrix. This maximum equals the actual entropy only for that thermal state and measures entanglement only for a specified subsystem of a pure state. The geometric multiplicity entropy follows if all emitted partons share one mode. An entropy computed from densities becomes such a maximum, an upper bound on the actual entropy, only for specified resolved modes. We also show that the state averaged over unrecorded emissions, assumed incoherent across scales, obeys probability-preserving Lindblad evolution on finitely many regulated modes with the resolution scale if emission and no-emission terms match. The occupations reproduce leading-order DGLAP, with virtual terms, at leading logarithmic accuracy as the cutoffs on momentum fractions are removed. The reduction holds for strongly ordered transverse momenta in light-cone gauge and dilute occupations, including those of newly resolved modes. Approximating momentum and helicity modes by independent thermal populations closes the occupation equation and gives this bound's evolution, with Bose and Pauli factors from the operator algebra. We propose testing this approximation at HERA or the LHC. The test compares the mode number inferred from normalized hadron factorial moments in a rapidity window with the window's transverse mode count.