# What is meant by the “Leading Log Approximation” in the context of the steep rise in gluon distribution at small bjorken x?

I am taking a course in QCD, and we're learning about Regge Theory, Pomeron exchange and structure functions.

I want to know what is meant by the leading log approximation in the context of the gluon distribution function. I take it has something to do with a very steep rise observed in the $$F_2(x, Q^2)$$ structure function, whereas Regge theory would predict a levelling off of $$F_2$$ at small x.

What I think I Know

• The optical theorem gives the cross section for some deep inelastic scattering process $$ab \to X$$: $$\sigma(ab \to X) = \sum_i C^{ab} S_{ab}^{\alpha_i - 1}$$, where $$C^{ab}$$ is a constant and $$S_{ab}^{\alpha_i -1}$$ is the centre of mass energy squared to the power $$\alpha_i -1$$. $$\alpha_i$$ is again a constant. The sum is over all possible virtual exchanges in the DIS process.
• Since bjorken $$x = Q^2/S$$, small $$x$$ corresponds to large $$S$$
• This means the cross section for the virtual exchange is related to $$x$$ via: $$\sigma(V^*p\to X) \propto x^{1-\alpha_i}$$
• Regge Theory predicts mesons and baryon spins are proportional to $$M^2$$. They can be plotted on a linear trajectory of spin vs $$M^2$$ , and the y-intercept of each line gives $$\alpha_i$$
• Pomeron exchange dominates at small $$x$$, with $$\alpha_{pom} = 0.08$$. The cross section is : $$\sigma(V^*p \to X) \propto x^{1-0.08} = x^{0.92}$$
• Since $$x^{0.92} \approx x^1$$, it was predicted that the structure function $$F_2(x, Q^2)$$ should level off at small $$x$$.
• This levelling off was not observed - in fact, it spikes up rapidly at small $$x$$ (demonstrated in HERA experimental data).

My Question

Our professor then said this spike is due to the gluon distribution, and it has problems for QCD calculations done using the leading log approximation.

• What does any of this have to do with gluons? How do gluons come into the $$F_2(x, Q^2)$$ structure function?
• What is the leading log approximation? I have seen this as a power series expansion of log terms, but unsure what is being expanded here, or why.