# Form factors in QCD pole

1. Oct 17, 2012

### 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.

2. Oct 17, 2012

### 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.

3. Oct 17, 2012

### 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, qμ is the 4-momentum transfer, the form factors do not have poles for spacelike, physical values of Q². The above mentioned equation should read

$$G(Q^2) \sim \frac{1}{1+Q^2/\mu^2}$$

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 = (2mπ)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

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