Homework 7

1) The most general form of the nucleon electromagnetic current is:

$$\Gamma_\mu = \Gamma_1(Q^2) \gamma_\mu + \Gamma_2(Q^2) \frac{i}{2} \sigma_{\mu\nu}q^\nu + \Gamma_3(Q^2) q_\mu$$

Demonstrate that because of current conservation, $\Gamma_3(Q^2) =0$.

2)

(a) What is the analytic form of the charge density distribution as a function of $r$ for the dipole form factor:

$$G_D(Q^2) = \frac{1}{\left(1+\frac{Q^2}{\alpha^2} \right)^2} ?$$

(Show your work).

(b) Express the root mean square radius, $\langle r^2 \rangle^{1/2}$ as a function of the parameter $\alpha$.

(c) Demonstrate that the derivative of the form factor as $Q^2 \rightarrow 0$ is proportional to $\langle r^2 \rangle$. What kind of physical information do we obtain from this?

(d) Under what hypothesis can we assume $\mid {\bf q} \mid \simeq \sqrt{Q^2}$?