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$\pi N$ coupling constant

Another motivation for a better direct determination of the pion-nucleon scattering lengths is connected to the $\pi N$ coupling constant $f_{\pi N}$. The Goldberger-Miyazawa-Oehme (GMO) sum rule [18], which is obtained from a forward dispersion relation for the isospin-odd $\pi N$ scattering amplitude, provides a relation between the scattering length $a^-$ and the $\pi N$ coupling constant $f_{\pi N}^2$:

$\displaystyle (1+\frac{m_{\pi}}{M})\;\frac{a^- }{m_{\pi}}$ $\textstyle =$ $\displaystyle \frac{2f_{\pi N}^2}{m_{\pi}^{2}-(m_{\pi}^2/2M)^2} + J$ (4)
$\displaystyle J$ $\textstyle =$ $\displaystyle \frac{1}{2\pi^2} \int_{0}^{\infty}\;\frac{\sigma^-(k)}{\omega(k)}\;dk \approx -0.053 m_{\pi}^{-2}$ (5)

Here $k$ and $\omega(k)$ are the incident pion momentum and energy, and $\sigma^-(k)=(\sigma^{tot}(\pi^-p)-\sigma^{tot}(\pi^+p))/2$ is the isospin-odd total cross section. The integral (5) is known with an accuracy of the order of 1% (see [36,37,38] and references therein). Therefore the determination of the width $\Gamma_{1S}(\pi^-p)$ with one percent accuracy offers the possibility to determine the pion-nucleon coupling with an accuracy of better than 1%. An immediate application is an evaluation of the Goldberger-Treiman discrepancy [39,40] which accounts for a small correction to the Goldberger-Treiman relation
$\displaystyle f_{\pi N}^2$ $\textstyle =$ $\displaystyle \frac{m_{\pi} g_A^2}{16\pi F_{\pi}^2} = 0.072$ (6)

where $g_A$ is the axial-vector coupling constant and $F_{\pi}$ is the weak pion decay constant.

next up previous contents
Next: Electromagnetic corrections. Up: Strong interaction Previous: Scattering length   Contents
Pionic Hydrogen Collaboration
1998