Constraining Dimension-Six Nonminimal Lorentz-Violating Electron--Nucleon Interactions with EDM Physics [CPT'19]

TL;DR

This paper investigates how dimension-six Lorentz-violating electron--nucleon interactions can induce electric dipole moments in atoms, and sets experimental bounds on these violations using EDM measurements.

Contribution

It classifies possible Lorentz-violating couplings in electron--nucleon interactions and constrains their magnitudes through EDM experimental data.

Findings

Constraints on Lorentz-violating coefficients at levels of 3.2×10⁻³¹ (eV)⁻² and 1.6×10⁻³³ (eV)⁻².

Identification of couplings compatible with electric dipole moment physics.

Analysis of the behavior of Lorentz-violating tensors under C, P, and T transformations.

Abstract

Electric dipole moments of atoms can arise from P-odd and T-odd electron--nucleon couplings. This work studies a general class of dimension-six electron--nucleon interactions mediated by Lorentz-violating tensors of ranks ranging from $1$ to $4$. The possible couplings are listed as well as their behavior under C, P, and T, allowing us to select the couplings compatible with electric-dipole-moment physics. The unsuppressed contributions of these couplings to the atom's hamiltonian can be read as equivalent to an electric dipole moment. The Lorentz-violating coefficients' magnitudes are limited using electric-dipole-moment measurements at the levels of $3.2\times10^{-31}\text{(eV)}^{-2}$ or $1.6\times10^{-33}\text{(eV)}^{-2}$.

Equations

$$\alpha^{i}=\left(\begin{array}[c]{cc}0&\sigma^{i}\\ \sigma^{i}&0\end{array}\right),\ \ \Sigma^{k}=\left(\begin{array}[c]{cc}\sigma^{k}&0\\ 0&\sigma^{k}\end{array}\right),\ \ \gamma^{0}=\left(\begin{array}[c]{cc}1&0\\ 0&-1\end{array}\right),\ \ \gamma^{5}=\left(\begin{array}[c]{cc}0&1\\ 1&0\end{array}\right),$$

$$\frac{\Delta E}{E_z}=2e\Re \left[⟨{\psi }_0|H_{P,T}|\eta ⟩\right]\equiv d_{\text{equiv}},$$

$$\left(\psi_{0}\right)^{l}=\left(\begin{array}[c]{c}\frac{i}{r}G_{l,J=\frac{1}{2}}(r)\phi_{\frac{1}{2},\frac{1}{2}}^{l}\\ \frac{1}{r}F_{l,J=\frac{1}{2}}(r)\boldsymbol{\sigma}\cdot\hat{\boldsymbol{r}}\phi_{\frac{1}{2},\frac{1}{2}}^{l}\end{array}\right),\ \eta^{l}=\left(\begin{array}[c]{c}\frac{i}{r}G_{l,J=\frac{1}{2}}^{S}(r)\phi_{\frac{1}{2},\frac{1}{2}}^{l}\\ \frac{1}{r}F_{l,J=\frac{1}{2}}^{S}(r)\boldsymbol{\sigma}\cdot\hat{\boldsymbol{r}}\phi_{\frac{1}{2},\frac{1}{2}}^{l}\end{array}\right),$$

Full text

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1 Introduction

Electric dipole moments (EDMs) are excellent probes for violations of discrete symmetries and for physics beyond the Standard Model.[1] EDM terms violate both parity (P) and time-reversal (T) symmetry, while preserving charge conjugation (C), assuming the CPT theorem holds. An atom’s EDM could arise from intrinsic properties of the electrons and/or nucleus, or from P- and T-odd electron–nucleon ($e$–$N$) couplings. EDM phenomenology can also arise in a Lorentz-violation (LV) scenario addressed within the framework of the Standard-Model Extension (SME).[2] LV generates CP violation and EDMs via radiative corrections[3] or at tree level via dimension-five nonminimal couplings,[4, 5] and may also yield nuclear-EDM corrections to the Schiff moment.[6] Nonminimal couplings have been of great interest in recent years.[7] In this work, we investigate a class of dimension-six LV $e$–$N$ couplings, composed of rank-$1$ to rank-$4$ background tensors, first proposed in Ref. \refciteLVArbDim, and the possible generation of atomic EDMs.[9]

2 Nonminimal $e$–$N$ Lorentz-violating couplings

The simplest couplings involve a rank-$1$ LV tensor $(k_{XX})_{\mu}$ with an effective lagrangian of the form $\mathcal{L}_{\text{LV}}=(k_{XX})_{\mu}\left[\left(\bar{N}\,\Gamma_{1}\,N\right)\left(\bar{\psi}\,\Gamma_{2}\,\psi\right)\right]^{\mu},$ where the upper $\mu$ belongs either to $\Gamma_{1}$ or $\Gamma_{2}$. The subscript $XX$ labels the type of fermion bilinear: scalar ($S$), pseudoscalar ($P$), vector ($V$), axial vector ($A$), and tensor ($T$), accounting for the $16$ linearly independent $4\times 4$ matrices. The operators $\Gamma_{1,2}$ must be combinations of Dirac matrices, given here as

[TABLE]

and $\gamma^{i}=\gamma^{0}\alpha^{i}$, $\sigma^{\mu\nu}=i\left[\gamma^{\mu},\gamma^{\nu}\right]/2$, with $\sigma^{0j}=i\alpha^{j}$, $\sigma^{ij}=\epsilon_{ijk}\Sigma^{k}$. As we are interested in EDM behavior, we select the P-odd and T-odd components. The rank-$1$ couplings are listed in Table 2, in which “S” and “NS” mean “suppressed” and “not suppressed,” respectively, in the nucleon’s nonrelativistic limit. Couplings of higher ranks are shown in Tables 2 to 2.

The EDM contribution, in the nonrelativistic limit for the nucleons, is calculated via atomic parity nonconservation methods[10] as

[TABLE]

where $H_{P,T}$ is the coupling’s hamiltonian contribution for the electron, and $|\eta\rangle,|\psi_{0}\rangle$—which have opposite parity—correspond to

[TABLE]

where $\phi_{\frac{1}{2},\frac{1}{2}}^{l=0}=(Y_{0}^{0},0)^{T}$ and $\phi_{\frac{1}{2},\frac{1}{2}}^{l=1}=(\sqrt{\frac{1}{3}}Y_{1}^{0},-\sqrt{\frac{2}{3}}Y_{1}^{1})^{T}$. While $\left(\psi_{0}\right)^{l}$ obeys Dirac’s equation for a central potential, $\eta^{l}$ obeys Sternheimer’s equation.[10]

By evaluating $d_{\text{equiv}}$ for each coupling via Eq. (2) and performing a sidereal analysis,[5] one can set upper bounds on the LV coefficients using numerical estimates on the thallium atom and the experimental limit on the electron’s EDM.[11] A list of time-averaged upper bounds is given in Table 2. More information can be found in Ref. \refciteadded.

Acknowledgments

We are grateful to CAPES, CNPq, FCT Portugal, and FAPEMA.