New Sum Rule for the Nuclear Magnetic Polarizability

The well-known Levinger-Bethe photonuclear sum rule relates the strength of the photoexcitation of the giant dipole resonance in a nucleus to the number of nucleons in that nucleus. I extend this sum rule to the case of virtual photons and relate the size of the magnetic polarizability of a nucleus to the $Q^2$ slope of the transverse virtual photoabsorption cross section integrated over the energy in the nuclear range. I check this sum rule for the deuteron where necessary data is available, discuss possible applications and connection with other sum rules postulated in the literature.

Figure 1

Total photoabsorption cross section on lead in $\mu$barn as a function of energy. Data in the nuclear range (blue crosses) extend up to ${\nu }_{\max }\approx 30\mathrm{MeV}$ and are from Ref. [11]. Data above the pion production threshold ${\nu }_{\pi }$ (red open circles) are from [12, 13, 14, 15]. The vertical dashed lines display ${\nu }_{\max }$, ${\nu }_{\infty }$, and ${\nu }_{\pi }$, see text for further details.

Figure 2

The comparison of the old fit without the use of the sum rule, $f_T^{\mathrm{FSI}}$ (blue dashed curve) and the new fit using the sum rule, ${\tilde{f}}_T^{\mathrm{FSI}}$ (red solid curve), with the uncertainty of each fit indicated by the band of the respective color. The sum rule is indicated by the star. The shaded band shows the kinematical range that is covered by the existing $D(e,e^{\prime })pn$ data. The inset in the upper right corner magnifies the small values of $Q^2$ where the slope of the new fit function is fixed to reproduce the value of ${\beta }_M^d$. Data points correspond to experimental data sets analyzed in Ref. [17] (Refs. [35–42] of that article).