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arXiv:2312.05712 · Letters in High Energy Physics

Entanglement Entropy Distributions of a Muon Decay

Shanmuka Shivashankara

Published Jan 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Divergences that occur in density matrices of decay and scattering processes are shown to be regularized by tracing and unitarity or the optical theorem.These divergences are regularized by the lifetime of the decaying particle or the total scattering cross section.Also, these regularizations are shown to give the expected helicities of final particles.As an illustration, the density matrix is derived for the weak decay of a polarized muon at rest, µ -→ ν µ (e -νe ), with Lorentz invariant density matrix entries and unitarity upheld at tree level.The electron's von Neumann entanglement entropy distributions are derived with respect to both the electron's emission angle and energy.The angular entropy distribution peaks for an electron emitted backward with respect to the muon's polarization given a minimum volume regularization larger than the cube of the muon's Compton wavelength.The kinematic entropy distribution is maximal at half the muon's rest mass energy.These results are similar to the electron's angular and kinematic decay rate distributions.Both the density matrix and entanglement entropy can be cast in terms of either ratios of areas or volumes.

DOI
10.31526/lhep.2024.531

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Divergences that occur in density matrices of decay and scattering processes are shown to be regularized by tracing and unitarity or the optical theorem.These divergences are regularized by the lifetime of the decaying particle or the total scattering cross section.Also, these regularizations are shown to give the expected helicities of final particles.As an illustration, the density matrix is derived for the weak decay of a polarized muon at rest, µ -→ ν µ (e -νe ), with Lorentz invariant density matrix entries and unitarity upheld at tree level.The electron's von Neumann entanglement entropy distributions are derived with respect to both the electron's emission angle and energy.The angular entropy distribution peaks for an electron emitted backward with respect to the muon's polarization given a minimum volume regularization larger than the cube of the muon's Compton wavelength.The kinematic entropy distribution is maximal at half the muon's rest mass energy.These results are similar to the electron's angular and kinematic decay rate distributions.Both the density matrix and entanglement entropy can be cast in terms of either ratios of areas or volumes.
Regularized quantum information metrics are calculated for the scattering process e − e + → γ , Z → μ − μ + that has a witness photon entangled with the initial electron-positron state. Unitarity implies the correct regularization of divergences that appear in both the final density matrix and von Neumann entanglement entropies. The entropies are found to quantify uncertainty or randomness. The variation of information, entanglement entropy, and correlation between the muon’s and witness photon’s helicities are found to convey equivalent information. The magnitude of the muon’s expected helicity rises (falls) as the helicity entropy falls (rises). Area, or the scattering cross section, is a source of entropy for the muon’s helicity entropy and momentum entropy. The muon’s differential angular entropy distribution is similar to the differential angular cross section distribution, capturing the forward-backward asymmetry at high center-of-mass energies. Published by the American Physical Society 2024
Unitarity and the optical theorem are used to derive the reduced density matrices of Compton scattering in the presence of a witness particle. Two photons are initially entangled wherein one photon participates in Compton scattering, while the other is a witness, i.e., does not interact with the electron. Unitarity is shown to require that the entanglement entropy of the witness photon does not change after its entangled partner undergoes scattering. The final mutual information of the electron's and witness particle’s polarizations is shown to be nonzero for low-energy Compton scattering. This indicates that the two particles became correlated in spite of no direct interaction. Assuming an initial maximally entangled state, the change in entanglement entropy of the scattered photon’s polarization is calculated in terms of Stokes parameters. A common ratio of areas occurs in the final reduced density matrix elements, von Neumann entropies, Stokes parameter, and mutual information. This common ratio consists of the Thomson scattering cross-section and an accessible regularized scattering area.
We are interested in the entanglement entropy of the final state of two particles in an elastic scattering. One can expect that the entanglement is yielded by the interaction between the particles. Within an S-matrix theory, by using the partial wave expansion of the S-matrix elements for two-particle states, we calculate a density matrix in a high energy elastic scattering, and obtain the entanglement entropy formula, which is described by physical observables. In the derivation of this formula, there appear divergences due to the infinite volume of two-particle Hilbert space. In order to avoid such divergences, we suggest a volume-regularization. Then we concretely evaluate the entanglement entropy for proton-proton scattering by using experimental data.
Entanglement of the two scattered particles is expected to occur in elastic collisions, even at high energy where they are in competition with inelastic ones. We study how to evaluate quantitatively the corresponding entanglement entropy SEE{S}_{\mathrm{EE}}SEE​. For this sake, we regularize the divergences occurring in the formal derivation of SEE{S}_{\mathrm{EE}}SEE​ using a regularization procedure acting on the two-particle Hilbert space of final states. A quantitative application is performed in proton-proton collisions at collider energies, comparing the results of SEE{S}_{\mathrm{EE}}SEE​ with two different cutoffs and with a volume-regularization obtained by a prescription fixing the finite two-body Hilbert space volume. A significant entanglement is found which persists even at the highest available energies.
We construct the entanglement entropy of a general scattering process between one particle species (denoted as AAA) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle AAA's Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle AAA, SAS_ASA​, in terms of scattering probabilities and differential cross sections, assuming no more than one particle AAA is produced in the final state. SAS_ASA​ contains the information of both the total scattering probabilities (or cross sections) and the differential distributions --- the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution (S∣i⟩S|i\rangleS∣i⟩) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle AAA is subleading. We also obtain the n≥2n\geq2n≥2 Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.
We study the behavior of entanglement between different degrees of freedom of scattering fermions, based on an exemplary QED scattering process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$. The variation of entanglement entropy between two fermions from an initial state to the final state was computed, with respect to different entanglement between the ingoing particles. This variation of entanglement entropy is found to be proportional to an area quantity, the total cross section. We also study the spin-momentum and helicity-momentum entanglements within one particle in the aforementioned scattering process. The calculations of the relevant variations of mutual information in the same inertial frame reveals that, for a maximally entangled initial state, the scattering between the particles does not affect the degree of both of these entanglements of one particle in the final state. It is also found that the increasing degree of entanglement between two ingoing particles would restrict the generation of entanglement between spin (helicity) and momentum of one outgoing particle. And the entanglement between spin and momentum within one particle in the final state is shown to always be stronger than that for helicity-momentum for a general initial entanglement state, implying significantly distinct properties of entanglement for the helicity and spin perceived by an inertial observer.