295 f(-p₁) + f(-p₂) → W^-(→ e^-(p₃) + ν(p₄)) + γ(p₅)

295.1 Wγ production, processes 290-299, 2941, 2991

These processes represent the production of a W boson which subsequently decays leptonically, in association with a real photon. Since this process includes 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 W → ℓν will include photon radiation from the lepton, so that a non-zero R(γ,ℓ)_min (Rgalmin) should also be supplied. This will ensure that the cross section is well-defined. Virtual amplitudes are taken from ref. [1].

The calculation of processes 290 and 295 may be performed at NLO and NNLO using the Frixione algorithm [2] or standard isolation. Thec phenomenology of these processes at NNLO has been treated in ref. [3].

For processes 290 and 295 the role of mtrans34cut changes to become a cut on the transverse mass on the M₃₄₅ system, i.e. the photon is included with the leptons in the cut.

Processes 292 and 297 represent the Wγ+jet production processes. They may be computed to NLO.

Processes 294 and 299 represent the photon-induced reactions, p + γ → eνγj and should be computed at NLO. Processes 2941 and 2991 represent the photon-induced reactions, p + γ → eνγjj and should be computed at NLO.

295.1.1 Anomalous WWγ couplings

Processes 290-297 may also be computed including the effect of anomalous WWγ couplings, making use of the amplitudes calculated in Ref. [4]. Including only dimension 6 operators or less and demanding gauge, C and CP invariance gives the general form of the anomalous vertex [4],

Γ αβμ (q,¯q,p) = ¯qαgβμ(2+ Δ κγ + λγ-q2-)- qβgαμ(2+ Δ κγ + λγ-¯q2-) W W γ M 2W M 2W μ μ αβ 1 2-λγ- -λγ- α β + (¯q - q )[- g (1 + 2p M 2 )+ M 2 p p ], (1) W W

where the overall coupling has been chosen to be -|e|. The parameters that specify the anomalous couplings, Δκ^γ and λ^γ, are specified in the input file as already discussed in Section ??. If the input file contains a negative value for the form-factor scale Λ then no suppression factors are applied to the anomalous couplings. Otherwise, the couplings are included in MCFM only after suppression by dipole form factors,

γ Δ κγ1 γ Δλγ Δ κ → (1+-ˆs∕Λ2)2, λ → (1+-ˆs∕Λ2)2,

(2)

where ŝ is the Wγ pair invariant mass.

The Standard Model cross section is obtained by setting Δκ^γ = λ^γ = 0.

295.2 Input files for non-local subtraction ref. [5]

295.3 Plotter

nplotter_Vgamma.f is the default plotting routine.

295.4 Example input and output file(s)

input295.ini process295.out

References

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

[3] J.M. Campbell, G. De Laurentis, R.K. Ellis and S. Seth, The pp → W(→ lν) + γ process at next-to-next-to-leading order, JHEP 07 (2021) 079 [2105.00954].

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

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