Axial Anomaly, entanglement and polarization

TL;DR

The paper investigates how the axial anomaly induces quantum entanglement between photon polarizations in pion decay and related high-energy processes, including time-like and space-like configurations relevant to heavy-ion collisions. It derives the VVP correlator $M^{\mu \nu} = C(q^2)\varepsilon^{\mu \nu \alpha \beta}k_\alpha p_\beta$ and the pseudoscalar-to-two-photon amplitude $M^{P\to \gamma\gamma} = C q^2 \varepsilon^{e_1^* e_2^*}$, showing that $|M|^2 = N(1 - \mathrm{Tr}(\rho_1 \rho_2))$ encodes entanglement via density-matrix relations $\rho_{ij}=e_i e_j^*$ and $\xi_{i,1}^f = -\xi_{i,2}$, $\xi_{i,2}^f = -\xi_{i,1}$. The analysis extends to the crossed $t$-channel case, where $|M|^2 = N(1 - \mathrm{Tr}(\rho_1^T \rho_2))$ yields $ ho_2^f = I - \rho_1$ and polarization-transfer patterns, illustrating retrodictive EPR-AB features. The work also connects these quantum correlations to classical-field analogues and to external magnetic-field effects that generate longitudinal polarization of time-like photons and vector mesons, supported by lattice studies of vacuum conductivity (DCME) and related phenomena. Overall, the paper provides a covariant entanglement framework linking axial anomaly, polarization observables, and heavy-ion phenomenology with potential implications for fundamental causality notions and spin-related sum rules.

Abstract

The (pion) decays controlled by axial anomaly imply the specific entanglement between photons having also the counterparts for classical electromagnetic waves. This is also a specific case of Eisnstein-Podolsky-Rosen-Bohm-Aharonov effect. The absence of causality and non-locality in (angular) momentum conservation is manifested, being especially clear for the generalization to the case of time rather than space separation corresponds to the polarization of dileptons described by time-like pion transition formfactors which may be studied experimentally. The similar decays in external magnetic field manifest the interplay with vacuum conductivity in external magnetic field and longitudinal polarization of vector mesons observed in heavy-ion collisions.