Constraints on the photon polarisation in $b \to s \gamma$ transitions using $B_s^0 \rightarrow \phi e^+e^-$ decays

An angular analysis of the $B_s^0 \rightarrow \phi e^+e^-$ decay is performed using the proton-proton collision dataset collected between 2011 and 2018 by the LHCb experiment, corresponding to an integrated luminosity of $9\,{\rm fb}^{-1}$ at centre-of-mass energies of 7, 8 and $13\,{\rm TeV}$. The analysis is performed in the very low dielectron invariant mass-squared region between $0.0009$ and $0.2615\,{\rm GeV}^2\!/c^4$. The longitudinal polarisation fraction of the $\phi$ meson is measured to be less than $11.5\%$ at $90\%$ confidence level. The $A_{\mathrm{T}}^{\mathcal{R}e C\!P}$ observable, which is related to the lepton forward-backward asymmetry, is measured to be $0.116 \pm 0.155 \pm 0.006$, where the first uncertainty is statistical and the second systematic. The transverse asymmetries, $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathcal{I}m C\!P}$ , which are sensitive to the virtual photon polarisation, are found to be $-0.045 \pm 0.235 \pm 0.014$ and $0.002 \pm 0.247 \pm 0.016$, respectively. The results are consistent with Standard Model predictions.