Nucleon form factors in dispersively improved chiral effective field theory: Scalar form factor

We propose a method for calculating the nucleon form factors (FFs) of $G$-parity-even operators by combining chiral effective field theory ($\chi \mathrm{EFT}$) and dispersion analysis. The FFs are expressed as dispersive integrals over the two-pion cut at $t>4M_{\pi }^2$. The spectral functions are obtained from the elastic unitarity condition and expressed as products of the complex $\pi \pi \to N\overline{N}$ partial-wave amplitudes and the timelike pion FF. $\chi \mathrm{EFT}$ is used to calculate the ratio of the partial-wave amplitudes and the pion FF, which is real and free of $\pi \pi$ rescattering in the $t$ channel ($N/D$ method). The rescattering effects are then incorporated by multiplying with the squared modulus of the empirical pion FF. The procedure results in a marked improvement compared to conventional $\chi \mathrm{EFT}$ calculations of the spectral functions. We apply the method to the nucleon scalar FF and compute the scalar spectral function, the scalar radius, the $t$-dependent FF, and the Cheng-Dashen discrepancy. Higher-order chiral corrections are estimated through the $\pi N$ low-energy constants. Results are in excellent agreement with dispersion-theoretical calculations. We elaborate several other interesting aspects of our method. The results show proper scaling behavior in the large-$N_c$ limit of QCD because the $\chi \mathrm{EFT}$ calculation includes $N$ and $\mathrm{\Delta }$ intermediate states. The squared modulus of the timelike pion FF required by our method can be extracted from lattice QCD calculations of vacuum correlation functions of the operator at large Euclidean distances. Our method can be applied to the nucleon FFs of other operators of interest, such as the isovector-vector current, the energy-momentum tensor, and twist-2 QCD operators (moments of generalized parton distributions).

Figure 1

Distribution of strength in the dispersive integrals for the scalar charge radius, Eq. (7) (solid red line), and the Cheng-Dashen discrepancy, Eq. (8) (dashed blue line). The plot shows the integrands as functions of $t$, divided by the value of the integral, i.e., normalized to unit area under the curves.

Figure 2

(a) Unitarity relation for the imaginary part of the nucleon scalar FF on the two-pion cut, Eq. (12). (b) Real function $J_{+}^0\left(t\right)$, Eq. (16), defined as the ratio of the $\pi \pi \to N\overline{N}$ PWA and the pion FF. (c) Unitarity relation in terms of $J_{+}^0\left(t\right)$ and the squared modulus of the pion FF, Eq. (15).

Figure 3

(a) Analytic structure of the $\pi \pi \to N\overline{N}$ PWA. The function has a right-hand cut resulting from the $t$-channel $\pi \pi$ intermediate state and a left-hand cut resulting from $s$-channel intermediate states ($N,\mathrm{\Delta },\pi N$, ...). The phase of the PWA on the right-hand cut is the same as that of the pion FF. (b) Analytic structure of the pion FF. The function has a right-hand cut resulting from the $\pi \pi$ intermediate state.

Figure 4

[(a), (b)] LO $\chi \mathrm{EFT}$ diagrams contributing to the $\pi \pi \to N\overline{N}$ PWA. (c) Pion scalar FF in LO.

Figure 5

LO $\chi \mathrm{EFT}$ result for the function $[3k_{\mathrm{cm}}/\left(4{\tilde{p}}_N^2\sqrt{t}\right)]J_{+}^0\left(t\right)$, which enters in the manifestly real unitarity relation Eq. (15). Dashed blue line: Contribution of $N$ Born term, Fig. 4. Solid red line: Sum of $N$ and $\mathrm{\Delta }$ Born terms, Figs. 4 and 4.

Figure 6

Adjustment of the LECs of the NLO $\pi \pi NN$ contact term in our unitarity-based approach. The original contact term (filled circle, left-hand side) is equated with the sum of $\sigma$ meson exchange and a reduced contact term (open circle). The reduced contact terms are used in the present scalar FF calculation with explicit unitarization.

Figure 7

Pion scalar FF obtained in the dispersive analysis of Ref. [35]. The plot shows the normalized squared modulus of the FF, $|{\sigma }_{\pi }{\left(t\right)|}^2/M_{\pi }^4$, as it enters in the improvement formula Eq. (17).

Figure 8

$\chi \mathrm{EFT}$ results for the function $3k_{\mathrm{cm}}/\left(4{\tilde{p}}_N^2\sqrt{t}\right)J_{+}^0\left(t\right)$, which enters in the manifestly real unitarity relation Eq. (15). Long-dashed red line: LO. Blue band with short-dashed contours: NLO. Red band with solid contours: NLO + N2LO, estimated as described in Sec. 2c. (The bands labeled NLO and NLO+N2LO show the total result up to that order and include the LO contribution.) Dash-dotted black line: Dispersion-theoretical result of Ref. [70].

Figure 9

$\mathrm{DI}\chi \mathrm{EFT}$ results for the scalar spectral function, Eq. (17). The upper plot (a) covers the full range up to $t=0.8{\mathrm{GeV}}^2$; the lower plot (b) covers the near-threshold region. The long-dashed red lines (LO approximation), blue bands with short-dashed contours (NLO), and red bands with solid contours (NLO+N2LO) correspond to those of Fig. 8. Dash-dotted black lines: Roy-Steiner result of Ref. [19].

Figure 10

Real part of the nucleon scalar FF $\mathrm{\Delta }\sigma \left(t\right)=\sigma \left(t\right)-\sigma \left(0\right)$, obtained from the once-subtracted dispersion integral Eq. (4) with the $\mathrm{DI}\chi \mathrm{EFT}$ spectral functions. The upper plot (a) shows the full $t$-range up to $0.5{\mathrm{GeV}}^2$, the lower plot (b) the near-threshold region. The long-dashed red lines (LO approximation), blue bands with short-dashed contours (NLO), and red bands with solid contours (NLO+N2LO) correspond to those of Figs. 8 and 9. Green bands with dotted contours: Dispersion-theoretical result of Ref. [17].

Figure 11

Two-pion cut of the scalar correlation function, Eq. (21).