Measuring Electron Energy in Muon-to-Electron Conversion using Holographic Synchrotron Radiation Emission Spectroscopy

The coherent conversion of a muon to an electron in a nuclear field has been one of the most powerful methods for searching for charged lepton flavor violation. Recent advancements have significantly enhanced the sensitivity of $\mu \rightarrow e$ searches, primarily driven by advancements in muon beamline design and low-mass tracking detectors, which afford exceptional momentum resolution. Nevertheless, the performance of these detectors is inherently limited by electron scattering and energy loss within detector materials. We propose a {\em holographic} track reconstruction method that leverages synchrotron radiation emitted by electrons to overcome these inevitable limitations. Similar to cyclotron radiation emission spectroscopy, which has demonstrated outstanding energy resolutions for low-energy electrons, our technique relies on a precision measurement of cyclotron frequency, but in a regime where photons are emitted stochastically and are projected onto a 2-dimensional inner surface of a solenoidal magnet. We outline the concept of such a massless holographic tracker and the feasibility of employing this innovative detection strategy for $\mu \rightarrow e$ conversion. We also address pertinent limitations and challenges inherent to the method.