Homework 3

1) By working out explicitely the components of the hadronic tensor in the parton model:

$$W_{\mu\nu}(P,q) = \sum_{i} \int d^4 k f_i(k,P) w_{\mu\nu}(q,k) \delta \left( (k+q)^2 \right),$$

with

$$w_{\mu\nu} = 2 k_\mu k_\nu + k_\mu q_\nu + q_\mu k_\nu - g_{\mu\nu} (kq)$$

,

we showed in class that the dimensionless structure function, $F_1$, can be written as:

$$F_1(x) = \sum_i e_i^2 \left[ \frac{pi}{2} \int d^2 k_T f_i(k,P) \right]$$

.

Derive using a similar procedure an expression for the structure function $F_2$.

2) Within the Infinite Momentum Frame discussed in class, derive an expression for the constant $C$ in the cross section:

$$d\sigma (\gamma^* k \rightarrow k^\prime) = C(x,Q^2) \delta(\xi-x),$$

where $\xi=k/P$, $k$ being the initial struck quark's momentum, $k^\prime$ being the quark's final momentum, $P$ being the proton's momentum, and $q$ the virtual photon momentum ($Q^2=-q^2$).