form factors in QCD pole

plasmon

Form factors in QCD are given in following form "f(q^2)=f(0)/(1-q^2/m^2)". This expression has a pole at q=m. How do we plot these form factors vs center of mass energy and deal with the pole.

andrien

I hope mass is still given a small negative imaginary part to deal with like qed propagators.May be it is not right with form factors.

tom.stoer

The electric (magnetic) form factor are measured via electron scattering i.e. via one-photon exchange between the electron and the charge (current) density of the hadron. Usually form factors are introduced as Fourier transform of the charge (current) density; but this picture is rather misleading when applied to non-perturbative QCD.

For Q² = -q² > 0, q^μ is the 4-momentum transfer, the form factors do not have poles for spacelike, physical values of Q². The above mentioned equation should read

[tex]G(Q^2) \sim \frac{1}{1+Q^2/\mu^2}[/tex]

The world data fit for μ² is 0.71 GeV/c². This is the standard dipole form factor; in reality the hadron form factors deviate from this simple form.

The definition of the form factors can be analytically continued to the complex q² plane. Here poles are excluded on the first q² sheet in the complex plane. It is expected that for timelike q² there is a complicated cut structure of a Riemann manifold in q² with multiple sheets. The first cut opens at q² = (2m_π)² which is the threshold for pion pair production.

Therefore the above mentioned dipole form factor is not realistic for Q² < 0. μ² = 0.71 GeV/c² is not related to a physical pole.

Remark: the form factors are expected to satisfy a dispersion relation like

[tex]G(q^2) = \frac{1}{\pi}\int^\infty_{(2m_\pi)^2}ds \frac{\text{Im}\,G(s)}{s-q^2}[/tex]