Compton Scattering Sum Rules and Their Saturation

Theoretical features of the many fixed-momentum-transfer dispersive sum rules which can be written for the 26 possible generalized nucleon Compton scattering amplitudes (retarded products of vector currents) are surveyed, and the sum rules are put to experimental test. Theoretical attention is focused on the occurrence of right-signature fixed poles in the angular momentum plane, such as the $j=1\mathrm{}$ fixed poles whose couplings are related to electromagnetic form factors by current algebra. Unitarity is used to estimate the sum-rule integrands in terms of data for the photoproduction processes $\gamma N\to \pi N$ and $\gamma N\to \pi \Delta$. Limitations of the data require that the sum rules be cut off at photon lab energy $E_{\mathrm{lab}}=1.12\mathrm{}$ GeV. The main results are as follows: (a) Reasonable evidence is presented that two-time-component current-algebra sum rules involving the electric and magnetic isovector form factors ${G_E}^V\mathrm{}\left(t\right)\mathrm{}$ and ${G_M}^V\mathrm{}\left(t\right)\mathrm{}$ are correct for small spacelike $-t$. If they are also to be correct for $-t\gtrsim -0.6\mathrm{} {\left(\frac{\mathrm{GeV}}{c}\right)}^2$, then the $\rho$ Regge pole must choose nonsense at $\alpha =0\mathrm{}$, and the associated wrong-signature fixed pole there must be multiplicative. A time-space current-algebra sum rule probably fails. (b) The separate isotopic components of the Drell-Hearn sum rule are investigated. Those with $I=0\mathrm{}$ exchange in the $t$ channel seem very successful, whereas the $I=1\mathrm{}$ exchange sum rule clearly fails. The failure indicates an important contribution of a hitherto unsuspected $J^P\left(I^G\right)=1^{+}\left(1^{-}\right)$ fixed pole. (c) Detailed results on wrong-signature antialgebra sum rules, on Regge-pole sum rules (FESR's), and on sum rules testing conspiracy are presented.