Quantum Calculations of the Cavity Shift in Electron Magnetic Moment Measurements

TL;DR

The paper provides the first fully quantum calculation of the cavity shift in electron $g-2$ measurements within a closed cavity. Using a mode-sum approach and contour integration, the authors renormalize the linearly divergent sums by subtracting the free-space contribution, obtaining results that exactly match the classical predictions for spherical and cylindrical cavities, while also offering a flexible framework to include systematic effects. They demonstrate consistency between nonrelativistic and relativistic treatments, highlighting the infrared nature of the cavity shift and its dependence on cavity geometry and mode structure. The approach lays groundwork for refining higher-precision measurements by enabling regulator-controlled extensions to more general cavities and imperfections, with direct relevance to upcoming electron $g-2$ experiments.

Abstract

The measurement of the anomalous electron magnetic moment $g-2$ through quantum transitions of a single trapped electron is the most stringent test of quantum field theory. These experiments are now so precise that they must account for the effects of the cavity containing the electron. Classical calculations of this "cavity shift" must subtract the electron's divergent self-field, and thus require knowledge of the exact Green's function for the cavity's electromagnetic field. We perform the first fully quantum calculation of the cavity shift in a closed cavity, which instead involves subtracting linearly divergent cavity mode sums and integrals. Using contour integration methods, we find perfect agreement with existing classical results for both spherical and cylindrical cavities, justifying their current use. Moreover, our mode-based results can be naturally generalized to account for systematic effects, necessary to push future measurements to the next order of magnitude in precision.

Figures (7)

• Figure 1: Schematic depiction of the technique for current measurements of electron $g-2$. Left: an electron is placed in a uniform magnetic field, which causes cyclotron motion and spin precession. A weak quadrupolar electrostatic field traps the particle in the axial direction parallel to $B$. Center: measuring the transition energies $\omega_a$ and $\omega_c$ allows $g-2$ to be extracted, independent of the magnitude of the $B$-field to leading order. Right: the entire system is placed in a cylindrical cavity which reflects the electron's cyclotron radiation, extending the cyclotron motion's lifetime and shifting the cyclotron frequency. This "cavity shift" is the focus of this work.
• Figure 2: One-loop contribution to the cavity shift in the relativistic quantum theory. The Schwinger propagator $S_A$ is the electron propagator in an external magnetic field, and the external electron states are approximate eigenstates of the full trapping potential, including the axial confinement.
• Figure 3: Integration contours, with poles marked with red crosses. Left: the original integration contour, in terms of $p$. Center: the same integration contour after changing variables to $c$. Right: moving the contour to the imaginary axis $C_v$ picks up half the residue of the pole at $c = 0$, and yields a pair of integrals, over $C_+$ and $C_-$, that can be related by symmetry.
• Figure 4: Integration contours, with poles marked with red crosses. We show the result for the part of the TM mode integral with a branch cut, though the TE mode integral is qualitatively similar. Left: the original integration contour, in terms of $n$. Center: an equivalent integration contour in terms of $c$, which diverts around the branch cut. Right: using symmetry, the two halves of the contour integral can be combined into the single contour $C_h+C_v$, which passes straight through poles on the real axis. The branch cut has been pushed downward for clarity. Alternatively, deforming the contour to $C_u$ yields an integral which is easier to numerically evaluate.
• Figure 5: Numeric values of the cavity shift for the cylinder, computed using our renormalized mode sum result Eq. \ref{['eq:cyl_answer']}, and using the existing classical Green's function result Eq. \ref{['eq:classical_cylinder']}. The two agree perfectly, and cannot be distinguished on the plot. The sums are evaluated up to $n = 50$, but summing to $n = 10$ already gives a match to $\sim 1\%$ accuracy. The top value of $L/a$ was chosen to match the most recent measurement Fan:2022oyb, and the bottom value was chosen to match that considered in Ref. PhysRevA.32.3204.