What kinds of contributions can be neglected in the leading logarithmic approximation? I'm looking for some good explanation on leading logarithmic approximation (LLA) in QCD; in particular, what types of contributions can be neglected while assuming LLA?
 A: Logarithmic series are a very broad topic. Generally speaking, many quantities in QCD can be expressed as power series of the form
$$X(s) = \underbrace{\sum_n X_{0n}(\alpha_s\ln s)^n}_\text{LL terms} + \underbrace{\sum_n X_{1n}\alpha_s(\alpha_s\ln s)^n}_\text{NLL terms} + \cdots$$
The kinds of contributions that enter each set of terms depend entirely on what quantity $X$ is being calculated.
I can only address this in more detail in the context of BFKL physics. The BFKL equation governs the unintegrated gluon distribution for a hadron, $\mathcal{F}$:
$$\mathcal{F}(\gamma) = \mathcal{F}^{(0)}(\gamma) + \frac{\bar{\alpha}_s}{\omega}\chi(\gamma)\mathcal{F}(\gamma) = \frac{\bar{\alpha}_s}{\omega}[\underbrace{\chi_0(\gamma)}_\text{LL} + \underbrace{\bar{\alpha}_s\chi_1(\gamma)}_\text{NLL} + \cdots]\mathcal{F}(\gamma)$$
where $\gamma$ is the Mellin conjugate to the momentum transfer $Q$. The best reference I've been able to find on what is included and excluded from the LL terms is this paper by Gavin Salam. He identifies three specific effects that contribute to the NLL and higher terms:


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*Running coupling
The running of the strong coupling (to one-loop order) is described by
$$\bar{\alpha}_s(Q) = \frac{\bar{\alpha}_s(Q_0)}{1 + b\bar{\alpha}_s(Q_0)\ln\frac{Q^2}{Q_0^2}}$$
At LL order, you can just use a constant $\bar{\alpha}_s(Q_0)$ for the coupling, but at NLL, you have to incorporate the fact that multiple energy scales are involved in the process, and the difference between $\bar{\alpha}_s$ at these multiple scales leads to NLL corrections.

*Splitting function
Part of the BFKL equation involves a splitting function which is associated with gluon branching ($1\to 2$) in the Feynman diagrams. The leading term in the splitting function is $\frac{1}{z}$, where $z$ is the fraction of momentum taken by one of the final-state gluons. That's sufficient for the LL expression, but when additional terms of the splitting function are taken into account, they produce NLL and higher contributions.

*Energy scale dependence
This is one effect that is more general than just BFKL physics. Recall that when we write $\alpha_s\ln s$ we actually mean $\alpha_s\ln\frac{s}{s_0}$, where $s_0$ is some arbitrary constant. Writing the same quantity for different values of $s_0$ involves differences which are at NLL and higher order.
But of course, those are just the most prominent, specifically identified terms. There are some other, smaller NLL corrections, and of course it's largely unknown what contributions come in at NNLL and higher order, so it's impossible to provide a complete list.
