Charge symmetry breaking and parity violating electron-proton scattering

Charge symmetry breaking contributions to the proton's neutral weak form factors must be understood in order for future measurements of parity violating electron-proton scattering to be definitively interpreted as evidence of proton strangeness. We calculate these charge symmetry breaking form factor contributions using chiral perturbation theory with resonance saturation estimates for unknown low-energy constants. The uncertainty of the leading-order resonance prediction is reduced by incorporating nuclear physics constraints. Higher-order contributions are investigated through phenomenological vertex form factors. We predict that charge symmetry breaking form factor contributions are an order of magnitude larger than expected from naïve dimensional analysis but are still an order of magnitude smaller than current experimental bounds on proton strangeness. This is consistent with previous calculations using chiral perturbation theory with resonance saturation.

Figure 1

The leading CSB contributions to the proton's neutral weak form factors in chiral perturbation theory. CSB effects arise in the pion loop diagrams (a) and (b) from the proton-neutron mass difference. The crossed circle in diagram (c) represents a CSB nucleon-photon interaction arising from short distance interactions that contributes at the same expansion order in chiral perturbation theory. Wave function renormalization also gives a CSB contribution not shown.

Figure 2

CSB contributions to the proton's neutral weak form factors from tree-level resonance exchange. CSB effects arise from mixing between the isoscalar $\omega$ and isovector $\rho$. The mixing vertex ${\Theta }_{\rho \omega }$ is denoted by a crossed circle. These diagrams provide a resonance saturation estimate for the unconstrained contact interaction shown in Fig. 1.

Figure 3

The CSB form factors (a) $G_M^{\text{CSB}}\left(Q^2\right)$ and (b) $G_E^{\text{CSB}}\left(Q^2\right)$ contributing to the proton's neutral weak form factors. The darker blue shaded regions show our leading-order prediction for the form factors. The unconstrained contact term ${\kappa }_{\text{CT}}^{\text{CSB}}$ is estimated with resonance saturation and the coupling choice $g_{\omega }=10$ incorporates the nuclear scattering constraints described in the text and in Table 1 from Refs. [15, 16]. The width of the shaded regions results from uncertainty in the resonance parameters ${\kappa }_{\omega }$ and ${\Theta }_{\rho \omega }$. The lighter gray shaded regions show the same results with the coupling $g_{\omega }=42$ taken by KL from dispersion analysis. The dotted red lines show the loop contributions only (no contact terms) calculated in RB$\chi$PT, and the dashed red lines show the loop contributions when phenomenological $\pi \pi \gamma$ and $\pi NN$ vertex form factors are included.

Figure 4

The CSB form factor contributions including uncertainty estimates for resonance parameters and for contributions beyond LO. The resonance parameters chosen correspond to an upper limit on $\rho \text{−}\omega$ mixing contributions consistent with nuclear scattering and binding energy measurements as described in the main text. The darker blue regions in panels (a) and (c) include only resonance uncertainty. The lighter red regions in panels (b) and (c) additionally include an estimate of higher-order-term uncertainty as characterized by phenomenological vertex form factors. This estimate is found by considering $G^{\text{CSB}}\pm \left|\Delta G^{\text{CSB}}\right|$, where $\Delta G^{\text{CSB}}$ is the difference between the CSB form factors with and without phenomenological vertex form factors. This estimate of higher-order uncertainty adds significant uncertainty to $G_M^{\text{CSB}}$ but negligible uncertainty to $G_E^{\text{CSB}}$.

Figure 5

The CSB form factor contributions (using coupling constants of Refs. [15, 16]) including uncertainty estimates for resonance parameters and for contributions beyond LO. The darker blue regions in panels (a) and (c) include only resonance uncertainty and are identical to the blue regions in Fig. 3. The lighter red regions in panels (b) and (c) additionally include an estimate of higher-order-term uncertainty as characterized by phenomenological vertex form factors as above. Finally, panel (d) shows only the loop contributions to $G_E^{\text{CSB}}$. These provide a leading-order $\chi$PT prediction for $G_E^{\text{CSB}}$ that is numerically dominated by the formally next-to-leading-order resonance contributions included in panel (c).