rules in general. The Drell–Hearn sum rule and the Adler–Weisberger sum rule are expected to be true in QCD. As we shall see later in section \[subsec:wsrope\], the Weinberg sum rules are in fact theorems of QCD in the chiral limit where the light quark masses are neglected.

### The Drell–Hearn Sum Rule

The sum rule in question [@DH66] relates the proton anomalous magnetic moment $F_{2}(0)$ to an integral over the energy of the difference of total photoabsorption polarized cross–sections $$\left[F_{2}(0)\right]^2 =
\frac{M_{{\mbox{\scriptsize\rm p}}}^2}{2\pi^2\alpha}\int_{\nu_{\mbox{\rm
th}}} ^{\infty}\frac{d\nu}{\nu}\left[\sigma_{\gamma{\mbox{\rm
p}}}^{\uparrow\uparrow} (\nu)-\sigma_{\gamma{\mbox{\rm
p}}}^{\uparrow\downarrow}(\nu)\right].$$ Here, $\sigma_{\gamma{\mbox{\scriptsize\rm
p}}}^{\uparrow\uparrow} (\nu) \left[
\sigma_{\gamma{\mbox{\scriptsize\rm p}}}^{\uparrow\downarrow}(\nu)\right]$ denote the total cross–section for the absorption of a circularly polarized photon of laboratory energy $\nu$ by a proton polarized with its spin parallel (antiparallel) to the photon spin, $M_{{\mbox{\scriptsize\rm p}}}$ is the proton mass and $\alpha\simeq 1/137$ the fine structure constant. The integral starts at the inelastic threshold $\nu_{{\mbox{\rm th}}}=m_{\pi}^2/2M_{\mbox{\scriptsize\rm p}}$.

The three ingredients which are needed to derive this sum rule are: [*dispersion relations*]{} for forward Compton–scattering, [*the optical theorem*]{} which relates total cross–sections to the imaginary part of forward scattering amplitudes, and a [*low–energy theorem*]{} which fixes the value of the real forward scattering amplitude at zero energy. We shall see 