In a series of papers, we have been producing ultraprecise predictions for photon, W and Z production at large transverse momentum using Soft Collinear Effective Theory .The spectrum of these gauge bosons is known at next-to-leading order in perturbation theory. The NNLO calculation is extremely difficult, and people have been working on it for 20 years. Instead of producing the exact result at NNLO, we approximate the NNLO by including the dominant terms. We can in fact include the dominant terms to all orders in pertubation theory, using the renormalization group.

In JHEP 1002 (2010) 040 (arXiv:0911.0681) by Thomas Becher and Matthew D. Schwartz, a factorization theory for direct photon production was derived in the threshold limit. The distribution looks like

A phenomenological analysis along with a comparison to Tevatron data was included in this paper.

When compared to NLO, the theory is a liitle low (red below data)

When compared to SCET, there is excellend agreement, and the uncetainty is smaller. The green curve (NNLL+NLO) agrees with the data at high pT where the calculation in SCET is accurate.

In (arXiv:1106.4310) by Thomas Becher, Christain Lorentzen and Matthew D. Schwartz, the same method was applied to W and Z bosons. Here the extra scale makes the calculation a bit more complicated.

One important issue in these calculations is the choice of scale setting. In fixed order calculations, there is no way to determine the proper scale except by educated guesswork. With effective theory, there are multiple modes which occur at different scales. These scales can at least be given a exact definition: choose the scale which minimizes scale sensitivity when only one of the hard, jet or soft functions is included.

 

Different modes choose their own scales -- here jet and hard. In the fixed order, NLO, there is no natural scale

Here the points are the minima of the curves on the left.

The minima are well approximated for various pT, S, and boson mass mV by the following equations

These choices correspond to the solid lines in the "Scale choices" plot above. For more examples of masses and energies to see the fit is good, click here for Jet Scale or Hard Scale.

More scale setting plots for Hard or Jet scales.