Constrained $\gamma Z$ interference corrections to parity-violating electron scattering

We present a comprehensive analysis of $\gamma Z$ interference corrections to the weak charge of the proton measured in parity-violating electron scattering, including a survey of existing models and a critical analysis of their uncertainties. Constraints from parton distributions in the deep-inelastic region, together with new data on parity-violating electron scattering in the resonance region, result in significantly smaller uncertainties on the corrections compared to previous estimates. At the kinematics of the $Q_{\mathrm{weak}}$ experiment, we determine the $\gamma Z$ box correction to be $\Re e{\square }_{\gamma Z}^V=(5.57\pm 0.36)\times {10}^{-3}$. The new constraints also allow precise predictions to be made for parity-violating deep-inelastic asymmetries on the deuteron.

Figure 1

Interference $\gamma Z$ box (left) and crossed box (right) diagrams. The wavy and dashed lines represent the exchanged $\gamma$ and $Z$ bosons, with the electron, hadron and virtual photon momenta labeled by $k$, $p$, and $q$, respectively.

Figure 2

Kinematic regions contributing to the ${\square }_{\gamma Z}^V$ integrals in the AJM model: Region I (blue shaded) at low $W$ and low $Q^2$ is described by the CB $F_{1,2}^{\gamma \gamma }$ fit [24], transformed to the $\gamma Z$ case; Region II (red shaded) represents the high-$W$, low-$Q^2$ domain as in Ref. 29 (or the GHRM Model II [14]), transformed to $\gamma Z$; and Region III (green shaded) at high $W$ and high $Q^2$ is described by global PDF fits to high-energy scattering data [40].

Figure 3

Proton $F_2^{\gamma \gamma }$ structure function versus $W^2$ at fixed $Q^2=0.05$, 0.5, 1.5 and $2{\mathrm{GeV}}^2$ for the CB fit [24] at low $W$ (blue solid) and $\mathrm{VMD}+\mathrm{\text{Regge}}$ parametrization [29] at high $W$ (red dashed). The boundary between these (corresponding to Regions I and II in Fig. 2) is indicated by the vertical dashed line at $W^2=9{\mathrm{GeV}}^2$.

Figure 4

Proton $F_2^{\gamma \gamma }$ structure function versus $W^2$ at fixed $Q^2=2.5$, 5, 8 and $10{\mathrm{GeV}}^2$ for the CB fit [24] at low $W$ (blue solid) and the ABM11 PDF parametrization [40] at high $W$ (green dotted), with the boundary between Regions I and III at $W^2=4{\mathrm{GeV}}^2$ indicated by the vertical line. For the $Q^2=2.5{\mathrm{GeV}}^2$ panel, the matching with the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model [29] (red dashed), corresponding to the boundary between Regions I and II, is indicated by the vertical line at $W^2=9{\mathrm{GeV}}^2$.

Figure 5

Proton $F_2^{\gamma \gamma }$ structure function versus $Q^2$ at fixed $W^2=4$, 6, 9 and $12{\mathrm{GeV}}^2$ for the CB fit [24] (blue solid), the ABM11 PDF parametrization [40] (green dotted), and the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model [29] (red dashed), with the boundaries between Regions I, II and III indicated by the vertical lines at fixed $Q^2$. Note that the small disagreement between the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model and the PDF parametrization for $Q^2=2.5{\mathrm{GeV}}^2$ appears only at larger $W^2$ values where the contribution to the dispersion integral is small.

Figure 6

Comparison of the proton $F_2^{\gamma Z}$ structure function in the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model (Model II) of GHRM [14] (red dashed) with the ABM11 global parametrization [40] (green dotted), for fixed $Q^2$ (top panels) and fixed $W^2$ (bottom panels). Note that the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model only includes uncertainties from the continuum part of the background, while the ABM11 parametrization includes an overall 5% error.

Figure 7

Continuum parameters ${\kappa }_C^T$ (left) and ${\kappa }_C^L$ (right) fitted to the DIS data, parametrized by the ABM11 global QCD fit [40], as a function of $W^2$ for fixed $Q^2=2.5{\mathrm{GeV}}^2$ (red triangles), $6{\mathrm{GeV}}^2$ (blue squares), and $10{\mathrm{GeV}}^2$ (green circles). The average values $⟨{\kappa }_C^{T,L}⟩$ are indicated by the solid lines, with the shaded band giving their uncertainty. Note that some of the points have been slightly offset for clarity.

Figure 8

Proton $F_2^{\gamma Z}$ structure function versus $W^2$ at various fixed $Q^2$ values for the low-$W$ CB fit [24] (blue solid) and the high-$W$ $\mathrm{VMD}+\mathrm{\text{Regge}}$ [29] (red dashed) and ABM11 [40] (green dotted) parametrizations. The boundaries between the Regions I, II and III are indicated by the vertical lines at $W^2=4$ and $9{\mathrm{GeV}}^2$.

Figure 9

Proton $F_2^{\gamma Z}$ structure function versus $Q^2$ at fixed $W^2=4$, 6, 9 and $12{\mathrm{GeV}}^2$ for the CB fit [24] (blue solid), the ABM11 PDF parametrization [40] (green dotted), and the $\mathrm{VMD}+\mathrm{\text{Regge}}$ model [29] (red dashed), with the boundaries between Regions I, II and III indicated by the vertical lines at fixed $Q^2$.

Figure 10

Proton parity-violating inelastic asymmetry $A_{\mathrm{PV}}/Q^2$ as a function of $W$, at fixed incident energy $E=0.69\mathrm{GeV}$ and $Q^2=0.34{\mathrm{GeV}}^2$, for the GHRM Model II [14] (left) and the AJM model (right). The data point at $W=1.18\mathrm{GeV}$ (black circle) is from the Jefferson Lab G0 experiment [53].

Figure 11

Proton parity-violating inelastic asymmetry $A_{\mathrm{PV}}/Q^2$ as a function of $W$, at fixed incident energy $E=6\mathrm{GeV}$ and $Q^2=2.5{\mathrm{GeV}}^2$, for the GHRM Model II [14] (left) and the AJM model (right). The asymmetry computed directly from PDFs [40] is represented by the green band.

Figure 12

Deuteron parity-violating asymmetry $A_{\mathrm{PV}}^d/Q^2$ as a function of $W$ for incident electron energy $E=4.9\mathrm{GeV}$ (left) and $E=6.1\mathrm{GeV}$ (right). The data points from the Jefferson Lab E08-011 experiment [15] at $W=1.26$ (green square), 1.59 (red circle), 1.86 (blue triangle) and 1.98 GeV (black diamond) correspond to average values of $Q^2=0.95$, 0.83, 0.76 and $1.47{\mathrm{GeV}}^2$, respectively. The AJM model uncertainties (inner dashed band) are constrained by matching the continuum parameters ${\kappa }_C^{T,L}(d)$ to the DIS region $\gamma Z$ structure functions [40], and are compared with those computed with errors on ${\kappa }_C^{T,L}(d)$ of 100% (outer dotted bands) and 25% (inner dotted bands).

Figure 13

As in Fig. 12, but with the AJM model asymmetries (solid) and their uncertainties (dashed) constrained by the E08-011 data [15]. Note the different scale on the ordinate to that in Fig. 12.

Figure 14

Energy dependence of the contributions to $\Re e{\square }_{\gamma Z}^V$ from the various regions in $W$ and $Q^2$ displayed in Fig. 2 in the AJM model (top), and the breakdown of Region I into its resonant and nonresonant background components (bottom).

Figure 15

Predictions for the parity-violating deuteron asymmetry $A_{\mathrm{PV}}^d/Q^2$ as a function of $W$ (solid) for the DIS region kinematics of the Jefferson Lab E08-011 experiment [54] at $Q^2=1.28{\mathrm{GeV}}^2$ (green), $1.09{\mathrm{GeV}}^2$ (red) and $1.90{\mathrm{GeV}}^2$ (blue) (see also Table ). The uncertainties (dashed) are computed in the AJM model with the continuum parameters ${\kappa }_C^{T,L}$ constrained by DIS structure functions (left), and by the E08-011 resonance region data (right). The predictions at the experimental $W$ values [54] are shown as pseudo-data points (open symbols).

Figure 16

AJM model prediction for the proton parity-violating asymmetry $A_{\mathrm{PV}}^p$ as a function of $W$ for the $Q_{\mathrm{weak}}$ inelastic measurement [16] at $Q^2=0.09{\mathrm{GeV}}^2$ (solid line and open symbol). The AJM model uncertainties (dashed) are compared with those from the GHRM model with 100% uncertainty on the continuum parameters (dotted).