# What is NLO?

This is not the place for a pedagogical explanation of field theory and perturbative calculations. That will come with the lectures.

First, one should understand that we calculate scattering probabilities based on a perturbative expansion in a small quantity. In fixed-order QCD calculations, we expand in the strong coupling at large momentum transfer, g_s. Since Feynman diagrams are squared, we see g_s**2/4pi=alpha_s. For Electroweak calculations, we expand in g_em, with g_em**2/4pi=alpha_em.

The leading or lowest order (or Born level) calculation is the one with the smallest power of the coupling that is possible. For q q~ -> W production, LO is 1 power of alpha_em and 0 powers of alpha_s.

Next-to-leading order (NLO) is one power of the coupling beyond LO. For q q~ -> W production, NLO in QCD is 1 power of alpha_em and 1 power of alpha_s. Since alpha_s >> alpha_em, the QCD correction is considered first.

With 1 power of alpha_s, there are 2 types of Feynman diagrams to consider.

1. Real diagrams, with a real gluon or quark in the final state.
2. Virtual diagrams, with a virtual gluon.

Virtual diagrams lead to infinities when the virtual gluon comes close to going on the mass shell. The real diagrams become infinite when an emitted gluon becomes soft (low energy) or collinear (parallel to one of the quarks). Fortunately, the two sets of infinities can be cancelled in such a way to yield a finite result.

# Why NLO?

The main virtue of NLO over LO is that the answer is:

1. More accurate: kinematic features begin to arise that cannot occur at LO. For example, the W boson has no transverse momentum at LO, but potentially large pT at NLO.
2. More stable: perturbative calculations in field theory require the introduction of an arbitrary momentum scale. It can be argued that this scale should be fixed by the process at hand. For example, the W mass M_W is a reasonable scale choice for W production. But is the best choice M_W or 2*M_W or 1/pi M_W? What does best choice mean? At NLO, the answer is much more stable to variations of the "arbitrary" scale. Formally, one can show the scale variation depends on one higher power of alpha. Since the scale variation is so reduced, the prediction of the normalization (i.e. the cross section) is more accurate.

# How to use NLO?

NLO Monte Carlo tools, as described here, are not event generators. Since they require a cancellation between real and virtual diagrams, it is not even possible to ask if there are exactly zero gluons in the final state. However, when suitably averaged, useful distributions can be made.

A technical point is that NLO calculations are made using weighted Monte Carlo methods. In practice, cross section weights come out positive and negative. If one chooses a suitable observable, and generates enough events, the positive and negative weights will cancel in a histogram, yielding a physical prediction.

# Why beyond NLO?

The next order in perturbation theory is called NNLO, for next-to-next-to-leading order. For Electroweak processes, NLO is the first time that alpha_s appears in the problem, so one cannot be confident that the result is converging. For these cases, NNLO is desired to study the scale variation. Indeed, for those processes calculated at NNLO, the scale variation is minimal (with the exception of Higgs boson production).