62 f(-p1) + f(-p2) W+(ν(p3) + μ+(p4)) + W-(q(p5) + q(p6))

62.1 Di-boson production, processes 61–89

These processes represent the production of a diboson pair V 1V 2, where V 1 and V 2 may be either a W or Z∕γ*. All the processes in this section may be calculated at NLO with the exception of nproc=69. There are various possibilities for the subsequent decay of the bosons, as specified in the sections below. Amplitudes for the V 1V 2 process at O(αs) are taken from ref. [1]. We also include singly resonant diagrams at NLO for all processes in the case zerowidth = .false.. For more details on this calculation, please see Refs. [23].

For processes 62, 63, 64, 65, 74 and 75 the default behaviour is that the hadronic decay products of the bosons are clustered into jets using the supplied jet algorithm parameters, but no cut is applied on the number of jets. This behaviour can be altered by changing the value of the variable notag in the file src/User/setnotag.f.

Calculations of processes 61, 71, 76, 81 and 82 can be performed at NLO by subtraction, zero-jettiness slicing and qT-slicing. They can be computed at NNLO using zero-jettiness slicing and qT-slicing, as described in ’Non-local slicing approaches for NNLO QCD in MCFM’,[4]. For processes 61, 81 and 86 the NNLO corrections include glue-glue initiated box diagrams which first contribute at order αs2. Two loop results for virtual diagrams at O(αs2) are taken from [5].

62.1.1 WW production, processes 61-65, 69

For WW production, both W’s can decay leptonically (nproc=61) or one may decay hadronically (nproc=62 for W- and nproc=64 for W+). Corresponding to processes 62,64, processes 63,65 implement radiation in decay from the hadronically decaying W’s. Process 69 implements the matrix elements for the leptonic decay of both W’s but where no polarization information is retained. It is included for the sake of comparison with other calculations. Processes 62 and 64 may be run at NLO with the option todk, including radiation in the decay of the hadronically decaying W. Processes 63 and 65 give the effect of radiation in the decay alone by taking the sum of the choices virt and real, or equivalently tota.

Note that, in processes 62 and 64, the NLO corrections include radiation from the hadronic decays of the W.

When removebr is true in processes 61 and 69, the W bosons do not decay.

Process 61 can be calculated at NNLO. The NNLO calculations include contributions from the process gg WW that proceeds through quark loops. The calculation of loops containing the third quark generation includes the effect of the top quark mass (but mb = 0), while the first two generations are considered massless. For numerical stability, a small cut on the transverse momentum of the W bosons is applied: pT(W) > 0.05 GeV for loops containing massless (first or second generation) quarks, pT(W) > 2 GeV for (t,b) loops. This typically removes less than 0.1% of the total cross section. The values of these cutoffs can be changed by editing src/WW/gg_ww.f and recompiling.

62.2 Plotter

nplotter_VV.f is the default plotting routine.

62.3 Example input and output file(s)

input62.ini process62.out

References

[1]    L.J. Dixon, Z. Kunszt and A. Signer, Helicity amplitudes for O(αs) production of W+W-, WZ, ZZ, Wγ, or pairs at hadron colliders, Nucl. Phys. B531 (1998) 3 [hep-ph/9803250].

[2]    J.M. Campbell and R.K. Ellis, An Update on vector boson pair production at hadron colliders, Phys. Rev. D 60 (1999) 113006 [hep-ph/9905386].

[3]    J.M. Campbell, R.K. Ellis and C. Williams, Vector boson pair production at the LHC, JHEP 07 (2011) 018 [1105.0020].

[4]    J.M. Campbell, R.K. Ellis and S. Seth, Non-local slicing approaches for NNLO QCD in MCFM, 2202.07738.

[5]    T. Gehrmann, A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for qqV 1V 2 4 leptons, JHEP 09 (2015) 128 [1503.04812].