Drell Hearn Sum Rule Research Articles

Spin Dependence of High-Energy Scattering Amplitudes. II. Zero-Mass Particles

The discussion of spin dependence at high energy is extended to the case of zero-mass particles. Crossing relations for zero-mass particles are derived, showing that the helicity of massless particles is simply reversed under crossing. These are used by way of a Pomeranchuk-Martin-type theorem to obtain restrictions on the high-energy spin dependence. The problem of fixed poles in the angular-momentum plane and the coupling of the Pomeranchuk trajectory to photons is discussed. It is shown that if there are no fixed poles at positive integers for any Compton amplitude, one obtains the (presumably ridiculous) result that the asymptotic total cross section for photons on any particle is proportional to the square of its charge. An approximate dynamical calculation is given which relates the coupling of photons to the Pomeranchuk trajectory with the derivative of the trajectory at $t=0$. This yields a prediction for the high-energy total cross section of photons on protons which is consistent with present data. The convergence of the Drell-Hearn sum rule is discussed; it is argued that even with cuts in the angular momentum, the sum rule will converge. Certain sum rules of B\'eg and of Pagels and Harari are discussed briefly.

Decaysπ0→2γ,η→2γand Sum Rules for Nucleon Compton Scattering

Motivated by the success of the Goldberger---Treiman calculation of charged-pion decay and the assumption of pion-pole dominance of the divergence of the axial current subsequently introduced, we study the decays of the neutral pion and eta in the same way. Examining Compton scattering of real photons from the nucleons presents a fruitful analogy to the nuclear $\ensuremath{\beta}$-decay process. Identifying the dominant nucleon and meson-pole-term contributions to Compton scattering and utilizing the content of the exact low-energy theorem on Compton scattering, a no-subtraction hypothesis, supported by the Regge-pole phenomenology, then enables us to establish exact sum rules for the lifetimes ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}$ and ${\ensuremath{\tau}}_{\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}}$ and also the Drell---Hearn sum rules for $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$. We also consider in detail the relation of our sum rule to the Goldberger---Treiman calculation of ${\ensuremath{\pi}}^{0}$ decay and forward-Compton-scattering sum rules for systems of spin=1. Neglecting continuum contributions, which are small, we find from our sum rules $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}=\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$ and $\ensuremath{\tau}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}^{}{}_{}{}^{\ensuremath{-}1}=\frac{\ensuremath{\pi}{\ensuremath{\alpha}}^{2}m_{\ensuremath{\pi}}^{}{}_{}{}^{3}\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}}{4{g}_{\ensuremath{\pi}{N}^{2}}{M}_{{N}^{2}}}\ensuremath{\simeq}3.1$ eV or ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}\ensuremath{\simeq}2.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ sec in approximate agreement with the experimental value ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}=(1.0\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ sec. Better photopion-production data for the nonresonant multipoles would enable us to accurately estimate the continuum contributions. Including the dominant contributions to the photopion production continuum, we find that the Drell-Hearn sum rule for $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}+\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$ is well satisfied. We can estimate the $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ lifetime but our result depends on a knowledge of the eta-nucleon coupling and on nonresonant background contributions to the photopion multipoles. Once better photopion production data for the nonresonant multipoles ${E}_{l\ifmmode\pm\else\textpm\fi{}}$, ${M}_{l\ifmmode\pm\else\textpm\fi{}}$, $ll~2$ for energies 1 BeV become available, these sum rules can offer reliable theoretical estimates for the decays ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$, $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$.