$N\pi$-state contamination in lattice calculations of the nucleon pseudoscalar form factor

The nucleon-pion-state contribution in the QCD three-point function of the pseudoscalar density is calculated to leading order in chiral perturbation theory. It predicts a nucleon-pion-state contamination in lattice estimates for the pseudoscalar form factor $G_{\mathrm{P}}(Q^2)$ determined with the plateau method. Depending on the momentum transfer $Q^2$ the contamination varies between $-20\%$ and $+50\%$ for a source-sink separation of 2 fm. The nucleon-pion-state contamination also causes violations in the generalized Goldberger-Treiman relation among the pseudoscalar and the axial nucleon form factors, the dominant source being the nucleon-pion-state contamination in the induced pseudoscalar form factor ${\tilde{G}}_{\mathrm{P}}(Q^2)$. Comparing the chiral perturbation theory predictions with lattice results of the PACS Collaboration we find reasonable agreement even for source-sink separations as small as 1.3 fm.

Figure 1

The deviation ${\epsilon }_{\mathrm{P}}^{\mathrm{mid}}(Q^2,t)$ for $t=2\mathrm{fm}$ as a function of $Q^2$ according to Eq. (4.1) with $M_{\pi }L=3$ (purple), 4 (blue), 5 (black) and 6 (red). Open symbols correspond to the results with the leading NR limit values for the coefficients, filled symbols include the $1/M_N$ corrections.

Figure 2

The relative deviations ${\epsilon }_{\mathrm{A}}^{\mathrm{mid}}(Q^2,t)$ (circles), ${\epsilon }_{\tilde{\mathrm{P}}}^{\mathrm{mid}}(Q^2,t)$ (diamonds) and ${\epsilon }_{\mathrm{P}}^{\mathrm{mid}}(Q^2,t)$ (triangles) defined in Eq. (4.4) as a function of $Q^2$ for $t=2\mathrm{fm}$. $Q^2$ values according to Eq. (4.1) with $M_{\pi }L=6$.

Figure 3

The ratios $r_1^{\mathrm{mid},\mathrm{N}\pi }(Q^2,t)$ (blue), $r_2^{\mathrm{mid},\mathrm{N}\pi }(Q^2,t)$ (red) and $r_{\mathrm{PCAC}}^{\mathrm{mid},\mathrm{N}\pi }(Q^2,t)$ (orange) given in Eqs. (4.7)–(4.9) for the lowest discrete momenta corresponding to $M_{\pi }L=6$ at $t=2\mathrm{fm}$ (filled symbols) and for infinite $t$ (open symbols).

Figure 4

The ratio $r_3^{\mathrm{mid},\mathrm{N}\pi }(Q^2,t)$, given in Eq. (4.12), for the lowest discrete momenta corresponding to $M_{\pi }L=6$ at $t=2\mathrm{fm}$ (filled symbols) and for infinite $t$ (open symbols).

Figure 5

The ratio $r_4^{\mathrm{mid},\mathrm{N}\pi }(Q^2,t)$, given in Eq. (4.13), for the lowest discrete momenta corresponding to $M_{\pi }L=6$ at $t=2\mathrm{fm}$ (filled symbols) and for infinite $t$ (open symbols).

Figure 6

PACS data for the normalized pseudoscalar form factor $G_{\mathrm{P}}^{\mathrm{norm}}(Q^2,Q_{\mathrm{ref}}^2,t)$, defined in Eq. (5.1), for $t=1.3\mathrm{fm}$ and $Q_{\mathrm{ref}}^2=0.072(\mathrm{GeV}{)}^2$ (black symbols). The dashed red line shows the PPD result. Red symbols correspond to the data corrected with Eq. (5.2).

Figure 7

PACS data (circles) and the ChPT result (diamonds) for the ratio $r_{\mathrm{PCAC}}^{\mathrm{plat}}(Q^2,t=1.3\mathrm{fm})$ given in Eqs. (4.7)–(4.9).

Figure 8

PACS data (circles) and the ChPT result (diamonds) for the ratio $r_3^{\mathrm{plat}}(Q^2,t=1.3\mathrm{fm})$ given in Eq. (4.12).

Figure 9

PACS data (circles) and the ChPT result (diamonds) for the ratio $r_4^{\mathrm{plat}}(Q^2,t=1.3\mathrm{fm})$ given in Eq. (4.13).