[Paper Review] CP violating phases in mu-e conversion

Sacha Davidson|ArXiv.org|Sep 1, 2008

Neutrino Physics Research11 references3 citations

TL;DR

This paper proposes that measuring the spin asymmetry of the electron in polarized muon-to-electron conversion on nuclei can probe CP-violating phases in lepton-flavor-violating dipole interactions. By leveraging the triple-product asymmetry involving muon and electron spins and electron momentum, the method detects CP violation through the imaginary part of the product $ A_L A_R^* $, offering sensitivity to new physics phases at future experiments with $ 10^{-16}-10^{-18} $ sensitivity.

ABSTRACT

Experiments are planned to improve the sensitivity of mu-e conversion from the current ~ 10^{-12} to 10^{-16} - 10^{-18}. If the muon (bound to the nucleus) could be polarised, a spin asymmetry of the final state electron is sensitive to CP violating phases on lepton flavour violating operators. This is similar to extracting phases from asymmetries in the final state spin and phase space distributions of mu to 3e and mu to e gamma.

Motivation & Objective

• To investigate whether CP-violating phases in lepton-flavor-violating dipole interactions can be probed via spin asymmetries in $\mu$-e conversion on nuclei.
• To demonstrate that polarized muons and measured electron polarization enable detection of CP violation through T-odd triple-product asymmetries.
• To establish a connection between CP-violating phases in the dipole operator coefficients $ A_L $ and $ A_R $, and measurable spin-dependent asymmetries in the final state.
• To provide a theoretical framework for extracting CP-violating phases in future high-sensitivity $\mu$-e conversion experiments with polarized muons.

Proposed method

• The study analyzes the matrix element squared for $\mu$-e conversion, focusing on the $ A_L A_R^* $ cross-term, which is sensitive to CP-violating phases.
• It uses Lorentz-invariant, CP-odd combinations of spin and momentum vectors, specifically $ \vec{s}_\mu \cdot (\vec{s}_e \times \vec{k}) $, to construct a triple-product asymmetry.
• The method applies polarization projection operators $ \frac{1}{2}(I + \gamma_5 \slashed{s}) $ to the initial and final state wavefunctions to account for polarized muons and electrons.
• The differential rate is derived by computing the matrix element squared with spin sums, isolating the $ A_L A_R^* $ term, which contributes to the asymmetry.
• The asymmetry is normalized to the total rate and expressed as a function of the imaginary part of $ A_L A_R^* $, scaled by the sum of $ |A_L|^2 + |A_R|^2 $.
• The analysis assumes that dipole interactions dominate over other dimension-six operators, and that the muon is polarized via the target, enabling spin-dependent measurements.

Experimental results

Research questions

• RQ1 Can CP-violating phases in lepton-flavor-violating dipole interactions be probed through spin asymmetries in $\mu$-e conversion?
• RQ2 What specific spin-momentum correlation is sensitive to the imaginary part of $ A_L A_R^* $, and how can it be measured?
• RQ3 How does the asymmetry depend on the relative magnitudes and phases of $ A_L $ and $ A_R $ in the dipole interaction?
• RQ4 What is the maximum achievable asymmetry for a given CP-violating phase, assuming polarized muons and measured electron polarization?

Key findings

• The spin asymmetry in $\mu$-e conversion is proportional to $ \text{Im}[A_L^* A_R] / (|A_L|^2 + |A_R|^2) $, providing a direct probe of CP-violating phases in the dipole interaction.
• The maximum asymmetry for 100% polarized muons is bounded by $ \frac{1}{8} \cdot \frac{\text{Im}[A_L^* A_R]}{|A_L|^2 + |A_R|^2} $, which is maximized when $ |A_L| \approx |A_R| $.
• The asymmetry arises from the triple-product term $ \vec{s}_\mu \cdot (\vec{s}_e \times \vec{k}) $, which is CP-odd and Lorentz-invariant.
• The method is sensitive to the imaginary part of $ A_L A_R^* $, which is not visible in the total branching ratio but appears in spin-dependent distributions.
• The analysis assumes that dipole interactions dominate, and that muons can be polarized via the target, enabling the measurement of electron polarization in the direction perpendicular to the muon spin and momentum plane.