Introduction

In deep inelastic scattering ( DIS) we probe the constituents of the proton with a virtual boson: $\gamma$, $Z^0$ or $W^{\pm}$. In the case of neutral boson exchange ($\gamma$, $Z^0$) we speak of Neutral Current ( NC) scattering, the exchange of a charged vector boson $W^+$ or $W^-$ is called Charged Current ( CC) scattering. Figure 2.1 shows the Feynman diagram for this process. We denote with $l$ ($l'$) the incoming (outgoing) lepton with four momenta $k$ ($k'$), $q \equiv (k - k')$, $P$ the initial state proton with four momentum $p$ and $H$ the hadronic final state.

Figure 2.1: Feynman diagram for deep inelastic lepton proton scattering.

It is conventional to describe the kinematics of the scattering with Lorentz scalars. The four momentum transfer

$${Q^2}\equiv -(k - k')^2 = -q^2$$ (1)

gives the length scale at which we probe

$$\lambda = 1/\sqrt{Q^2}$$ (2)

The maximum ${Q^2}$ is given by the center-of-mass energy squared of the lepton-proton system

$$s \equiv ( p + k )^2$$ (3)

Further the inelasticity,

$$y \equiv \frac{q \cdot p}{k \cdot p}$$ (4)

is, in the proton rest frame, the fraction of the energy transferred from the lepton to the struck quark.

Another convenient variable

$$x \equiv \frac{{Q^2}}{2 q \cdot p}$$ (5)

gives the fraction of the proton four momentum carried by the struck quark.

Only two of the variables ${x}$, ${Q^2}$ and ${y}$ are needed to fully describe the kinematics. For $s \gg {m_P}^2$ ($m_P$ is the proton mass) the following relation between the above quantities holds

$${Q^2}= x y s$$ (6)

The invariant mass of the hadronic final state is given by

$$W^2 = {Q^2}\frac{1-x}{x} + m_p^2$$ (7)

In terms of these variables the cross section for neutral current scattering is given by

$${\frac{d^2 \sigma_{\mbox{\em NC}}}{d x d Q^2}}(e^{\pm}p) = ... ...,Q^2) + (1 - y)F_2(x,Q^2) \mp (y - \frac{y^2}{2})x F_3(x,Q^2))$$ (8)

The structure functions $x F_1$, $F_2$ and $x F_3$ stem from general parameterization of the hadronic tensor [1].

For the naive quark parton model with massless quarks $2 x F_1 = F_2$ and the structure functions $F_2$ and $x F_3$ can be written as

$$F_2(x,Q^2) = \sum_q{A_q(Q^2)(x q(x) + x \bar{q}(x))}$$ (9)

$$xF_3(x,Q^2) = \sum_q{B_q(Q^2)(x q(x) - x \bar{q}(x))}$$ (10)

with $q(x), \bar{q}(x)$ being the quark and anti-quark densities in the proton respectively and $A_q(Q^2)$, $B_q(Q^2)$ describing the coupling of quark of flavor $q$ to the exchanged vector boson. $A_q(Q^2)$ can be written as

$$A_q(Q^2) = e_l^2 e_q^2 + 2 \vert e_l\vert \vert e_q\vert v_l v_q (\frac{1}{4 \sin^2 \theta_W \cos^2 \theta_W})(\frac{Q^2}{Q^2 + M_Z^2})$$

$$+ ( v_l^2 + a_l^2 )(v_q^2 + a_q^2) ( \frac{1}{4 \sin^2 \theta_W \cos^2 \theta_W})(\frac{Q^4}{(Q^2 + M_Z^2)^2})$$ (11)

with $e_l$, $v_l$, $a_l$ and $e_q$, $v_q$, $a_q$ the charge, vector and axial-vector coupling of the initial lepton and scattered quark respectively.

$B_q(Q^2)$ can be expressed as:

$$B_q(Q^2) = 2 \vert e_l\vert \vert e_q\vert a_l a_q (\frac{1}{4 \sin^2 \theta_W \cos^2 \theta_W}) (\frac{Q^2}{Q^2 + M_Z^2})$$

$$+ 4 v_l a_l v_q a_q ( \frac{1}{4 \sin^2 \theta_W \cos^2 \theta_W})(\frac{Q^4}{(Q^2 + M_Z^2)^2})$$ (12)

One can identify taking the first term of equation (2.8) and equations (2.11) and (2.12) the propagator terms for the exchange of a $\gamma$ $(1/Q^4)$, the $Z^0$ $(1/(Q^2 + M_Z^2)^2)$ and the interference of $Z^0$ and $\gamma$ exchange $(1/(Q^2(Q^2 + M_Z^2)))$.

For charged current scattering the cross section is:

$${\frac{d^2 \sigma_{\mbox{\em CC}}}{d x d Q^2}}(e^{\pm}p) = ... ... + M^2_W)^2} ((1 + (1-y)^2)W^{\pm}_2 \mp (1-(1-y)^2)W^{\pm}_3)$$ (13)

$W_2$ and $W_3$ are the sum and difference respectively of the quark and anti-quark densities. Since charge is conserved at the $W^{\pm} q$ vertex only those quarks which actually contribute to the cross section are taken:

$$W^{+}_2 = x \sum_i{(d_i(x) + \bar{u}_i(x))}$$

$$W^{-}_2 = x \sum_i{(u_i(x) + \bar{d}_i(x))}$$

$$W^{+}_3 = x \sum_i{(d_i(x) - \bar{u}_i(x))}$$

$$W^{-}_3 = x \sum_i{(u_i(x) - \bar{d}_i(x))}$$ (14)

Again one can identify in equation (2.13) the propagator term for the $W^{\pm}$ exchange $(1/(Q^2 + M^2_W)^2)$.

Clearly, when comparing the cross section for charged and neutral current scattering one can readily see that for relatively small ${Q^2}$ CC scattering is significantly less probable than NC scattering. The CC cross section is relatively flat for ${Q^2}$ up to the mass of the $W^{\pm}$ squared.

It is important to note that in the naive quark parton model the structure functions are independent of ${Q^2}$. When gluon radiation of the quarks and splitting of gluons into quark anti-quark pairs are taken into account ( QCD improved quark parton model), the parton density functions become indeed dependent on ${Q^2}$. The dependence is described in leading order QCD by the DGLAP [2] equations:

$$\frac{d q_f(x,Q^2)}{d log Q^2} = \frac{\alpha_s(Q^2)}{2 \pi} \int_z^1 ( q_f(z,Q^2) P_{qq}(\frac{x}{z}) + G(z,Q^2) P_{qg}(\frac{x}{z})) \frac{d z}{z}$$ (15)

$$\frac{d G(x,Q^2)}{d log Q^2} = \frac{\alpha_s(Q^2)}{2 \pi} \int_z^1 ( G(z,Q^2) P_{gg}(\frac{x}{z})$$

$$+ \sum_f (q_f(z,Q^2) + \bar{q}_f(z,Q^2)) P_{gq}(\frac{x}{z})) \frac{d z}{z}$$ (16)

Here $G(x,Q^2)$ is the gluon density in the proton and the ``splitting functions'' $P_{ij}(y)$ give the probability of obtaining a parton $i$ from parton $j$ where parton $i$ has a fraction $y$ of the momentum of parton $j$.

With these equations it is possible to calculate the parton density functions at all $Q^2$ if they are given at a certain $Q^2_0$. In next to leading order the equations become more cumbersome and the densities become dependent on the renormalization scheme. Within each scheme, however, the parton densities should be universal functions applicable to any interaction. In particular it should be possible to use parton density functions extracted from NC scattering to calculate CC scattering and visa versa.