What are Next-to-Leading-Order (NLO) corrections?

I'm currently studying Particle Physics and HEP and this acronym is omnipresent. I know it means next-to-leading-order but, what is exactly the physical meaning of LO and NLO?

• Have you learned about quantum field theory and Feynman diagrams yet? – David Z Apr 30 '18 at 9:25
• Just tree level – Enri Apr 30 '18 at 9:39
• This question is difficult to answer in its generality since you already know the only thing that is true in general: NLO means "next to leading order". The physical meaning of such contributions depends on what exactly you are computing, could you perhaps give a more concrete example you wish to have explained? – ACuriousMind Apr 30 '18 at 9:59

Now, the whole point of Feynman diagrams is to expand the phyisical quantity we are interested in as a power series, $$\sigma= a_0 + \alpha\cdot a_1 + \alpha^2 \cdot a_2 +\dotsm$$ For this to make sense, the expansion parameter $\alpha$, which usually is (related to) some coupling constant, needs to be small -- for example, in QED, $\alpha\approx 1/137$. Then, the higher-order terms, i.e. those with higher powers of $\alpha$, are suppressed, and the lower-order terms dominate (note that I'm glossing over quite a number of issues here to get the basic picture across).
In other words, the lowest-power of $\alpha$ with a nonzero term $a_i$ gives the most important contribution, called leading order, and the other ones are higher-order. In particular, the next one is the next-to-leading order (NLO), contribution and so on (for example, this paper computes NNNNLO contributions: https://arxiv.org/abs/hep-ph/0610143).
• Usually the series is in terms of $\alpha$ (because vertices occur in pairs once you square the amplitude). That's sort of tangential to your point but I wonder if it might be better to use $alpha$ as the expansion parameter instead of $g$ because that's what we usually see in practice. Just a minor thing and probably not worth an edit on its own. – David Z Apr 30 '18 at 22:23