These processes represent the production of a diboson pair V _{1}V _{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 _{1}V _{2} process 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. [2, 3].

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 q_{T}-slicing. They can be computed at NNLO
using zero-jettiness slicing and q_{T}-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
α_{s}^{2}.

The Z’s can either both decay leptonically (nproc=81), one can decay leptonically while the other decays into neutrinos (nproc=82) or bottom quarks (nproc=83), or one decays into neutrinos and the other into a bottom quark pair (nproc=84). In process 83 the mass of the b-quark is neglected. Note that, in processes 83–84, the NLO corrections do not include radiation from the bottom quarks that are produced by the Z decay. In process 90 the two Z bosons decay to identical charged leptons, and interference effects between the decay products of the two Z bosons are included. In all cases these processes also include the contribution from a virtual photon.

When removebr is true in process 81, neither of the Z bosons decays.

This process has been treated in several papers, [2, 3, 5, 4]. Processes 81 and 82
can be calculated at NNLO. The NNLO calculation includes contributions from the
process gg → ZZ that proceeds through quark loops. The calculation of loops
containing the third quark generation includes the effect of both the top and the
bottom quark mass (m_{t},m_{b}≠0), while the first two generations are considered
massless. For numerical stability, a small cut on the transverse momentum of the Z
bosons is applied: p_{T}(Z) > 0.1 GeV. This typically removes less than 0.1% of the
total cross section. The values of these cutoffs can be changed by editing
src/ZZ/getggZZamps.f and recompiling.

It is possible to specify anomalous trilinear couplings for the W^{+}W^{-}Z and W^{+}W^{-}γ
vertices that are relevant for WW and WZ production. To run in this mode, one
must set zerowidth equal to .true. and modify the appropriate lines for the
couplings in input.ini (see below). Note that, at present, the effect of anomalous
couplings is not included in the gluon-gluon initiated contributions to the WW
process.

The anomalous couplings appear in the Lagrangian, = _{SM} + _{anom} as follows
(where _{SM} represents the usual Standard Model Lagrangian and _{anom} is taken
from Ref. [6]):

where X_{μν} ≡ ∂_{μ}X_{ν} - ∂_{ν}X_{μ} and the overall coupling factors are g_{WWγ} = -e,
g_{WWZ} = -ecotθ_{w}. This is the most general Lagrangian that conserves C and P
separately and electromagnetic gauge invariance requires that there is no equivalent
of the Δg_{1}^{Z} term for the photon coupling.

In order to avoid a violation of unitarity, these couplings are often included only after suppression by dipole form factors,

where ŝ is the vector boson pair invariant mass and Λ is an additional parameter giving the scale of new physics, which should be in the TeV range. These form factors should be produced by the new physics associated with the anomalous couplings and this choice is somewhat arbitrary. The use of the form factors can be disabled as described below. The file input.ini contains the values of the 6 parameters which specify the anomalous couplings. If the input file contains a negative value for the form-factor scale then the suppression factors described above are not applied.

nplotter_ZZlept.f is the default plotting routine.

[1]
L.J. Dixon,
Z. Kunszt
and
A. Signer,
Helicity
amplitudes
for
O(α_{s})
production
of
W^{+}W^{-},
W^{±}Z,
ZZ,
W^{±}γ,
or
Zγ
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] R. Boughezal, J.M. Campbell, R.K. Ellis, C. Focke, W. Giele, X. Liu et al., Color singlet production at NNLO in MCFM, Eur. Phys. J. C 77 (2017) 7 [1605.08011].

[6]
L.J. Dixon,
Z. Kunszt
and
A. Signer,
Vector
boson
pair
production
in
hadronic
collisions
at
order
α_{s}
:
Lepton
correlations
and
anomalous
couplings,
Phys.
Rev.
D60
(1999)
114037
[hep-ph/9907305].