**W Helicity**

There are 8 helicity cross sections for the W boson, that determine, along with the total cross section, its angular coefficients in a hadron collider.

Here &theta and &phi are the angles of the charged lepton from the W decay in the Collins-Soper W rest frame
(the frame where the z-axis bisects the proton and minus the antiproton directions in the W rest frame).
The quantities q_{T} and &eta and are the transverse momentum and rapidity of the W in the lab frame.
Three of the coefficients (A_{5},A_{6},A_{7}) are next-to-leading-order (NLO) ones
and they appear only if gluon loops are included in the production of the W boson. These
NLO coefficients may be related to Parity-odd and Time-odd effects in the W production and
decay. To study the coefficients in high-pT, we need jets to balance the W transverse momentum.
These jets introduce QCD effects, that we would like to test.

Using a generic NLO in a_{s} generator (DYRAD), I extracted the SM prediction for
the 8 angular coefficients using the method of moments:

You need to zoom in to see A_{1} and the three next-to-leading order ones:

By dividing with the total unpolarized cross section, I calculated the 8 helicity cross sections of the W boson produced in hadron collisions:

In all these plots, the bands represent the energy scale and PDF systematic uncertainties.
** Three of these coefficients (A _{1},A_{4},A_{6}) and corresponding helicity
cross sections -- including a NLO one -- were presented for the ﬁrst time in literature **.
The results were published in PRD (Phys. Rev. D

**73**, 052001 (2006)).