Moments of nucleon structure functions at next-to-leading order in baryon chiral perturbation theory

We obtain leading- and next-to-leading-order predictions of chiral perturbation theory for several prominent moments of nucleon structure functions. These parameter-free results turn out to be in overall agreement with the available empirical information on nearly all of the considered moments, in the region of low momentum transfer $\left(Q^2<0.3{\mathrm{GeV}}^2\right)$. Especially surprising is the situation for the spin polarizability ${\delta }_{LT}$, which thus far was not reproducible in chiral perturbation theory for proton and neutron simultaneously. This problem, known as the “${\delta }_{LT}$ puzzle,” is not seen in the present calculation.

Figure 1

One-$\pi N$-loop graphs contributing to Compton scattering at $\mathcal{O}\left(p^3\right)$. Graphs obtained from these by crossing and time reversal are not shown, but are evaluated too.

Figure 2

Graphs contributing at $\mathcal{O}(p^4/\Delta )$. Double lines denote the propagator of the $\Delta$ isobar. Graphs obtained from these by crossing and time reversal are evaluated too.

Figure 3

Scalar polarizabilities of proton and neutron. Red solid lines and blue bands represent, respectively, the LO and NLO results of this work. Blue dashed line is the LO result in the HB limit. Black dotted lines represents the empirical result of MAID2007 [26]. The data points at $Q^2=0$ correspond with Refs. [27] and [28] (red and purple points, respectively) for the proton, and [28] for the neutron. The data point in the left upper panel at $Q^2=0.3$ ${\mathrm{GeV}}^2$ is from Ref. [29].

Figure 4

Generalized spin polarizabilities of proton and neutron. Red solid lines and blue bands represent, respectively, the LO and NLO results of this work. Black dotted lines represent MAID2007. Grey bands are the covariant $\mathrm{B}\chi \mathrm{PT}$ calculation of Ref. [30]. The blue dashed line is the $\mathcal{O}\left(p^4\right)$ HB calculation [31], off the scale in the upper panels. The red band is the IR calculation [32]. The data points for the proton ${\gamma }_0$ at finite $Q^2$ are from Ref. [7] (blue dots), and at $Q^2=0$ from [8] (purple square). For the neutron all the data are from Ref. [9].

Figure 5

Generalized GDH integral and inelastic part of the $d_2$ moment. The legend is the same as in the previous figure, except for the $\mathcal{O}\left(p^4\right)$ HB result (blue dashed line) which here is from Ref. [33], and the data points which are from Ref. [10] for $I_A$ and Ref. [11] for ${\overline{d}}_2$.

Figure 6

Left panel: $I_1=(2M_N^2/Q^2){\Gamma }_1$ for the proton. Right panel: isovector part of the Bjorken integral. Legend for the curves is as in Fig. 4. Data points for $I_1^p$ are from [7], for ${\Gamma }_1^{p-n}$ from [12] (squares) and [13] (dots).