Chiral corrections to the Adler-Weisberger sum rule

The Adler-Weisberger sum rule for the nucleon axial-vector charge, $g_A$, offers a unique signature of chiral symmetry and its breaking in QCD. Its derivation relies on both algebraic aspects of chiral symmetry, which guarantee the convergence of the sum rule, and dynamical aspects of chiral symmetry breaking—as exploited using chiral perturbation theory—which allow the rigorous inclusion of explicit chiral symmetry breaking effects due to light-quark masses. The original derivations obtained the sum rule in the chiral limit and, without the benefit of chiral perturbation theory, made various attempts at extrapolating to nonvanishing pion masses. In this paper, the leading, universal, chiral corrections to the chiral-limit sum rule are obtained. Using PDG data, a recent parametrization of the pion-nucleon total cross sections in the resonance region given by the SAID group, as well as recent Roy-Steiner equation determinations of subthreshold amplitudes, threshold parameters, and correlated low-energy constants, the Adler-Weisberger sum rule is confronted with experimental data. With uncertainty estimates associated with the cross-section parametrization, the Goldberger-Treimann discrepancy, and the truncation of the sum rule at $\mathcal{O}(M_{\pi }^4)$ in the chiral expansion, this work finds $g_A=1.248\pm 0.010\pm 0.007\pm 0.013$.

Figure 1

The SAID parametrization [5] superposed on the PDG data [27]. This SAID solution is used only in the resonance region (region II).

Figure 2

Parametrization of ${\sigma }^{-}$ in the threshold region with respect to the lab momentum, $k$, of the incoming pion. The dashed and solid lines correspond to S-wave and S/P-wave determinations of this quantity. Clearly, the P-wave is an essential contributor. The low-energy dashed region of the SAID WI08 observables solution has not been corrected for Coulomb effects and is thus replaced with the effective range expansion in this region.

Figure 3

Three distinct (Hendrick [31], Höhler [16], and PDG [27]) power-law fits to the high-energy region of the sum-rule integrand with respect to lab momentum of the incoming pion, $k$, as in Eq. (6.3). The dashed lines of corresponding color represent the lower limit to the parametrization’s claimed domain of validity. The sporadic black line (ending at the right, black, dashed vertical line) is a raw depiction of the PDG data through $\mathcal{O}(1)$ interpolation of the individual $\pi p$ cross sections contributing to $2{\sigma }^{-}$ as in the decomposition of Eq. (2.3).

Figure 4

Integrands expressed in the integration variable $k$ associated with the first two subthreshold coefficients. Note that $k$ is not the variable chosen to express these coefficients in Eqs. (4.3) and (4.4). Thus, the solid vertical line is placed at the $\mathrm{\Delta }(1232)$ contribution as a pertinent reference. For higher moments, the integrand tends to increase the influence of the threshold region as well as the influence of the $\mathrm{\Delta }(1232)$ relative to higher $\mathrm{\Delta }$ and $N$ resonances. This is consistent with the numerical findings that the value of subthreshold parameters in Table 2 are closely approximated when considering only the $\mathrm{\Delta }(1232)$ resonance.

Figure 5

Calculated value of $g_A$ with increasing upper bound of the integrated total cross section within the AW sum rule, Eq. (5.1). Results in the main plot are from the evaluation of Eq. (5.3) with the LEC values of Ref. [6]. The subplot includes the corresponding evaluation with Eq. (5.2) as well as two recent measurements of $g_A$ from neutron $\beta$-decay [33, 34] for comparison. The light gray line represents a similar analysis with the total cross section saturated by $\delta$-function resonances [35] and this paper’s evaluation of subthreshold parameters. The light-blue band is the 2012 PDG value for $g_A$.