Parity Nonconservation in Rb and Sr$^+$ due to Low-Mass Vector Boson

TL;DR

This work investigates parity-nonconservation (PNC) in light atoms as a probe for a hypothetical light vector boson $Z'$, extending prior analyses to rubidium and strontium ions. The authors compute the nuclear-spin-independent and -dependent PNC amplitudes $E_{\text{PNC}}^{a\to b}$ for the transitions $5s-6s$ and $5s-4d_{3/2}$ in Rb and Sr$^+$ using a relativistic all-order framework that incorporates core polarization, Breit-type corrections, and Brueckner orbitals, while treating the $Z'$-mediated interaction via a Yukawa potential and a Yukawa-type amplitude $A_{\rm PNC}^{Z'}$. They demonstrate that the $Z'$ contribution relative to the SM scales more favorably in lighter atoms and provide tables of $Z'$-induced amplitudes as a function of $m_{Z'}$, enabling extraction of constraints on $|g_e^A g_N^V|/m_{Z'}^2$ (for heavy $Z'$) or on $|g_e^A g_N^V|$ (for light $Z'$). With an optimistic relative deviation $\varepsilon=0.001$, the study projects about an order-of-magnitude improvement in limits for large $m_{Z'}$ and up to ~40× for small $m_{Z'}$ compared with Cs, highlighting the practical potential of light-atom PNC experiments in probing new physics. Key formulas include the PNC Hamiltonian $\hat{H}_{\text{PNC}}=\hat{H}_{\text{SI}}+\hat{H}_{\text{SD}}=\frac{G_F}{\sqrt{2}}(-\frac{Q_W}{2}\gamma_5+\frac{\varkappa}{I}\boldsymbol{\alpha}\cdot \boldsymbol{I})\rho(\boldsymbol{r})$, the vector-boson Yukawa potential $V_{12}(r)=\frac{g_1^A g_2^V}{4\pi}\frac{e^{-m_{Z'} r}}{r}\gamma_5$, and the amplitude relation $E_{\text{PNC}}^{a\to b}=\sum_n\left[\frac{\langle a|\hat{W}|n\rangle\langle n|\mathbf{D}|b\rangle}{E_a-E_n}+\frac{\langle a|\mathbf{D}|n\rangle\langle n|\hat{W}|b\rangle}{E_b-E_n}\right]$.

Abstract

We calculate the parity non-conserving (PNC) electric-dipole ($E1$) transition amplitudes for the $5s - 6s$ and $5s - 4d_{3/2}$ transitions in Rb and Sr$^+$. Our results include both the nuclear-spin-independent and nuclear-spin-dependent contributions, with particular emphasis on the potential effects of a hypothetical additional $Z'$-boson. We highlight possible advantages of using light atoms in searches for such new interaction. The ratio of the contribution of a low mass $Z'$-boson to the contribution of the Standard model $Z$-boson to PNC effects increases rapidly (faster than $1/Z^2$) with decreasing nuclear charge $Z$. Another advantage is that theoretical interpretations of experiments in lighter systems may be carried out with a higher accuracy than that in Cs, Ba$^+$, Fr and Ra$^+$.