Processes 300 and 305 represent the production of a Z boson (or virtual photon
for process 300) in association with a real photon based on ref. [1]. The
Z∕γ^{*} subsequently decays into either an e^{+}e^{-} pair (nproc=300) or neutrinos
(nproc=305). Since these processes include a real photon, the cross section
diverges when the photon is very soft or in the direction of the beam. Thus in
order to produce sensible results, the input file must supply values for both
gammptmin and gammrapmax. Moreover, when the parameters zerowidth
and removebr are set to .false. the decay Z → e^{-}e^{+} (nproc=300) will
include photon radiation from both leptons, so that a non-zero R(γ,ℓ)_{min}
(Rgalmin) should also be supplied. This will ensure that the cross section is
well-defined. The calculation of processes 300 may be performed at NNLO
using the Frixione algorithm [2] or standard isolation. The one-loop virtual
diagrams are taken from [3] and the two-loop virtual diagrams are taken from
[4].

For the process 300 the role of mtrans34cut changes to become a cut on the
invariant mass on the M_{345} system, i.e. the photon is included with the leptons in
the cut.

Processes 300-305 may also be computed including the effect of anomalous couplings between Z bosons and photons. Note that, at present, the effect of anomalous couplings is not included in the gluon-gluon initiated contributions.

The anomalous ZγZ vertex (not present at all in the Standard Model), considering operators up to dimension 8, is given by [5],

where the overall coupling has been chosen to be |e| (and ϵ^{0123} = +1). The
non-standard Z_{α}(q_{1})γ_{β}(q_{2})γ_{μ}(p) momentum-space vertex can be obtained from this
equation by setting q_{1}^{2} → 0 and replacing h_{i}^{Z} → h_{i}^{γ}. The parameters that specify
the anomalous couplings, h_{i}^{Z} and h_{i}^{γ} (for i = 1…4), are specified in the input file as,
e.g. h1(Z) and h1(gamma). If the input file contains a negative value for the
form-factor scale Λ then no suppression factors are applied to these anomalous
couplings. Otherwise, the couplings are included in MCFM only after suppression by
dipole form factors,

| (1) |

where ŝ is the Zγ pair invariant mass. Note that these form factors are slightly different from those discussed in Sections ?? and ??. The form factors can be modified in src/Need/set_anomcoup.f.

The Standard Model cross section is obtained by setting h_{i}^{Z} = h_{i}^{γ} = 0 for
i = 1…4.

nplotter_Vgamma.f is the default plotting routine.

[1] J.M. Campbell, T. Neumann and C. Williams, Zγ Production at NNLO Including Anomalous Couplings, JHEP 11 (2017) 150 [1708.02925].

[2] S. Frixione, Isolated photons in perturbative QCD, Phys. Lett. B429 (1998) 369 [hep-ph/9801442].

[3]
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].

[4]
T. Gehrmann
and
L. Tancredi,
Two-loop
QCD
helicity
amplitudes
for
q → W^{±}γ
and
q → Z^{0}γ,
JHEP
02
(2012)
004
[1112.1531].

[5] D. De Florian and A. Signer, W gamma and Z gamma production at hadron colliders, Eur. Phys. J. C16 (2000) 105 [hep-ph/0002138].

[6]
T. Becher
and
T. Neumann,
Fiducial
q_{T}
resummation
of
color-singlet
processes
at
N^{3}LL+NNLO,
JHEP
03
(2021)
199
[2009.11437].