300 f(-p1) + f(-p2) Z(e-(p3) + e+(p4)) + γ(p5)

300.1 , production, processes 300, 305

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 M345 system, i.e. the photon is included with the leptons in the cut.

300.1.1 Anomalous ZZγ and Zγγ couplings

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 ϵ0123 = +1). The non-standard Zα(q1)γβ(q2)γμ(p) momentum-space vertex can be obtained from this equation by setting q12 0 and replacing hiZ hiγ. The parameters that specify the anomalous couplings, hiZ and hiγ (for i = 14), 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,

 Z∕γ      hZ1∕,3γ         Z∕γ      hZ2∕,4γ
h1,3  → (1+-ˆs∕Λ2)3,    h2,4 →  (1+-ˆs∕Λ2)4,

where ŝ is the 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 hiZ = hiγ = 0 for i = 14.

300.2 Input file for transverse resummed cross-sections[6]

300.3 Plotter

nplotter_Vgamma.f is the default plotting routine.

300.4 Example input and output file(s)

input300.ini process300.out


[1]    J.M. Campbell, T. Neumann and C. Williams, 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(αs) production of W+W-, WZ, ZZ, Wγ, or pairs at hadron colliders, Nucl. Phys. B531 (1998) 3 [hep-ph/9803250].

[4]    T. Gehrmann and L. Tancredi, Two-loop QCD helicity amplitudes for qq Wγ and qq Z0γ, 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 qT resummation of color-singlet processes at N3LL+NNLO, JHEP 03 (2021) 199 [2009.11437].