Non-local slicing approaches for NNLO QCD in MCFM

We present the implementation of several processes at Next-to-Next-to Leading Order (NNLO) accuracy in QCD in the parton-level Monte Carlo program MCFM. The processes treated are $pp\to H$, $W^\pm$, $Z$, $W^\pm H$, $ZH$, $W^\pm\gamma$, $Z\gamma$ and $\gamma\gamma$ and, for the first time in the code, $W^+W^-$, $W^\pm Z$ and $ZZ$. Decays of the unstable bosons are fully included, resulting in a flexible fully differential Monte Carlo code. The NNLO corrections have been calculated using two non-local slicing approaches, isolating the doubly unresolved region by cutting on the zero-jettiness, ${\cal T}_0$, or on $q_T$, the transverse momentum of the colour singlet final-state particles. We find that for most, but not all processes the $q_T$ slicing method leads to smaller power corrections for equal computational burden.


Introduction
The current and future success of the LHC depends crucially on the precision supplied by theoretical calculations. Reducing the theoretical error has special importance in the context of Higgs Physics [2] where for many channels it is projected to remain the largest error at the conclusion of the LHC program.
An important role is played by processes involving the bosons, W, Z, γ and H. The importance of detailed studies of the Higgs boson goes without saying. Single vector boson production can be used as a luminosity monitor and to probe parton distribution functions. Electroweak production of vector boson pairs is a stringent test of the standard model. In addition the γγ, ZZ, W W and Zγ processes have renewed significance because they constitute backgrounds to Higgs boson decay processes.
We are preparing a release of MCFM [3][4][5][6] which will allow calculation of the NNLO QCD results for a large number of colour singlet production processes with both zerojettiness [7][8][9] and q T -slicing [10]. 1 The code has also been used for processes with noncolour singlet final states, such as W + jet [8], Z +jet [11], γ +jet [12], and Higgs boson + jet [13,14], although these have not yet been made available in the public version. These latter processes are treatable because of the SCET factorization theorems for the cross sections for small 1-jettiness.
Given the importance of precision for the LHC, there has been an intense community effort to produce results at NNLO QCD and in some cases N 3 LO. A review of the 2020 status of precision QCD with a special focus on the Higgs boson is given in Ref. [15]. In Table 1 we present references for the processes that have been calculated in NNLO QCD. It is important to note that when targeting the precision achievable at NNLO, electroweak corrections can also become important, especially at large p T . A discussion of these effects is beyond the scope of this paper.
considered totally independent, relying as they do on the same two-loop amplitudes [90,91]. We also compare with an calculation of ZZ production [1], which is independent (except for the same caveat about two-loop matrix elements).
Recently there has been a detailed re-examination of fiducial cross sections for two-body decay processes at colliders, demonstrating that certain commonly used cuts are sensitive to low momentum scales [92,93]. 2 Where possible we shall limit our discussion in this paper to total inclusive cross sections, leaving detailed predictions with well-motivated cuts to a subsequent paper.
A successive improvement of our results could come from widespread inclusion of resummation effects along the lines of Refs. [96,97]. Resummation effects in vector boson pair production have previously been considered in Refs. [98][99][100]. Resummation is also part of the program in the GENEVA collaboration. Ref. [101] provides a recent article, where references to earlier work of the GENEVA collaboration can be found.

Non-local slicing methods
In this section we review the calculation of the NNLO cross sections which have colour singlet final states at the Born level. The necessary requisites for the methods are, • An analytic understanding of the behaviour of the Born process accompanied by soft and collinear radiation through to the requisite order, i.e. for NNLO through to order α 2 s .
• A NLO calculation of the process at hand with one additional parton.
• The two-loop virtual corrections to the process at hand, necessary to calculate the hard function at order α 2 s Let r be a zero jet resolution variable which divides the phase space in two, where X represents other kinematics on the phase space. In the following subsections we shall take the resolution variable, r cut to be either the 0-jettiness, T 0 or the transverse momentum, q T of the final state.
The cross section is given by introducing σ sub , the analytic form for the cross section, known for small values of the resolution parameter r cut from factorization theorems.
Since r is a zero jet resolution variable, ∆σ(r cut ) will tend to zero as r cut → 0.

Non-local jettiness slicing
The 0-jettiness slicing method is based on the corresponding event shape introduced in Ref. [7]. Writing q µ , Q and Y for the four-momentum, mass and rapidity of the colour singlet system in its centre of mass, the incoming parton momenta are where n = (1, + z),n = (1, − z). The zero-jettiness in the colour singlet centre of mass is then defined by, where the sum over k runs over all final state partons and p − k = p.n, p + k = p.n. The allorders resummed form of the cross section in the region of small T 0 , obtained by application of soft-collinear effective theory (SCET) [102][103][104][105][106], is then given by, where the indices i, j run over all initial state partons involved in the scattering. Φ B represents the Born-level color singlet phase space p i p j → p singlet and H ij the hard function. The soft/collinear function ∆ ij is, The hard function encodes both the leading order matrix elements and perturbative virtual corrections as described later in section 2.3. The beam function B i/H describes initial-state collinear radiation from hadron H and can be written as a convolution of perturbative matching coefficients and the usual parton density functions, f i/H . It has been computed up to two loops in Refs. [107,108]. The effects of soft radiation are collected in the soft function S, which has been calculated for zero-jettiness up to two-loop order in Refs. [109,110].
In the color singlet centre of mass frame the power corrections to the factorization in Eq. (2.6) are known to be reduced [111]. Power corrections to the simplest 2 → 1 processes are known [111][112][113][114][115] but, since they are not known universally and we also wish to compare with the q T approach, we do not include them in this study.

Non-local q T slicing
In this section we briefly describe the calculation using the transverse momentum as a resolution parameter. Although the formalism we describe is not the formalism in which q T slicing was originally implemented [116][117][118][119][120][121] it is simplest to implement q T slicing using the factorized form of the low q T cross section derived using SCET. Schematically, the differential cross section takes the form, where the symbol ⊗ denotes a convolution. Note that the soft functionS and the naive transverse PDFsB depend on unphysical parameters, ξ 1 and ξ 2 . However, in physical cross sections the dependence on these parameters appears in such a way that only the physical scale Q remains. TheB andS functions still depend on both Q and q T , two disparate scales. As such, Eq. (2.8) does not represent a true factorization and thus additional work must be performed to isolate the dependence on the scale Q. We follow the SCET re-factorization approach of [122,123]. Thus for the simplest Drell-Yan process we have, which is correct up to power corrections in q 2 T /Q 2 and x 2 T Λ 2 QCD , where x 2 T = −x 2 ⊥ and x ⊥ is the Fourier conjugate variable to q T . The perturbative function C qq←ij is given in terms of the Wilson coefficient C V as, This is the form in which we have implemented the factorization formula, taking the explicit form for the functions I q/i and F from Ref. [124]. Since all the formula are explicitly given in a machine readable format this gives the simplest implementation method. An alternative approach would be to use Ref. [125] where a number of useful results are collected. Ref. [125] has the ambition to be more complete, since it also gives results which are relevant at N 3 LO, but for our purposes, i.e. NNLO, it is less useful since for some of the needed components it refers to other papers. We have checked that a full NNLO implementation based on Ref. [125] and papers referenced therein gives the same result as in Ref. [124]. The q T spectrum for color singlet production including the complete q 2 T /Q 2 power corrections at O(α s ), has been presented in ref. [126]. Power corrections for the NLO inclusive cross section of Drell-Yan processes have been calculated up to fourth order in a transverse-momentum cut in Ref. [127], and up to second order for the NNLO qg-initiated channel in Ref. [128].Therefore a full suite of power corrections at NNLO is not available, even for the simplest processes.

Hard functions
The hard functions are related to finite parts of the virtual one-and two-loop corrections to the Born process. For the case of the Drell-Yan type processes, (W, Z and γ * ) the two-loop corrections are given in Refs. [129,130]. For the case of Higgs production in the large m t limit the two-loop corrections are given in Refs. [130][131][132][133]. For the V γ (V = Z, W ± ) processes the finite remainders of the one-loop and two-loop form factors are given in [90], while the remainders for the γγ process are specified in Ref. [134]. For the diboson (W + W − , W ± Z, ZZ) processes, with the vector bosons decaying leptonically, the matrix elements up to two loops have been computed in Ref. [91]. Details of the conversion of the two-loop matrix elements in Ref. [91] to the hard functions are presented in Appendix B. We employ HandyG [135] for the numerical evaluation of multiple polylogarithms that appear in the expression of two-loop finite remainders.

Above cut contributions
A necessary ingredient for the vector boson pair calculations is the NLO calculation of the desired parton process but with one additional parton in the final state. In preparation for this paper we have implemented and improved the treatment of the V V + jet process at one-loop, for the cases of V V = W + W − , W ± Z, ZZ. As with all NLO processes in MCFM these are included using analytic formula. Analytic results for the one-loop calculation of the W + W − +3 parton process have been given in Ref. [136]. After simple modifications these results can also be applied to the W Z and ZZ processes. Although the results of Ref. [136] were in analytic form, considerable effort has been devoted to simplifying these results [137]. Analytic results for the one-loop calculation of the W + γ+3 parton process, applicable also to the Zγ process, have been given in Ref. [30].
We have compared our one-loop results with OpenLoops 2 [89] and Recola2 [138,139], both as a confirmation of the results and to establish timings. The comparison is performed via an extension of the C++ interface to MCFM [140], computing the interference of Born and 1-loop amplitudes using the same set of 1000 representative phase-space points with the default setup for both OpenLoops 2 (version 2.1.2) and Recola2 (version 2.2.3). Our calculations agree perfectly with those of these libraries 3 , with timing results shown in Table 2. Although the evaluation of this part of the full NNLO result is not the most expensive in terms of computing time, our results are in all cases faster than both Openloops 2 and Recola2.

Comparative study of jettiness and q T slicing
In this section we exploit the leading logarithmic dependence on the transverse momentum cut, q cut T and jettiness cut, τ cut , to define the appropriate variables to compare the two approaches. The leading logarithmic behaviour of a colour singlet cross section integrated Parton channel up to a small cutoff value, q cut T , is where σ 0 is the Born level cross section. The corresponding leading log formula for zerojettiness integrated up to a cut of value τ cut is, A simple derivation of these two formulas at order α s is given in Appendix A. The resources needed for a computation of a given accuracy is dominated by the calculation of the above-cut contribution. Comparing Eqs. (3.1) and (3.2) one therefore expects a similar size for the contribution coming from the above cut region when the values of τ cut and q cut T are related by [141], We therefore define the following two dimensionless quantities to encapsulate the slicing dependence of the results, and The computational burden is then expected to be very similar for equal values of T and τ and therefore we will compare the two schemes at the same values of T and τ . Although this argument is only made at the level of leading logarithms, we will see later (c.f. Table 5 in Section 3.4) that it is indeed supported even at NNLO for the operating values of τ and T that we choose. We note that all the results presented in this paper are obtained using a modified version of the MCFM-9.0 code, thus allowing the computation of cross sections at multiple values of T or τ in one run [6]. In order to present a fair comparison between the two approaches we generate the phase space in an identical way in both cases, one that has not been optimized for either.

Processes and cuts
For simplicity, and in order to avoid issues associated with the application of fiducial cuts in 2-body decays [92,93], we present results for inclusive Z, W ± , H, ZH and W ± H production. Decays of the W , Z and H bosons are not included and no cuts are applied. The remaining processes we compute are: As indicated, these processes include a full set of contributing Feynman diagrams and are not limited to those containing on-shell W or Z propagators. We will, however, often use these names as shorthand in the remainder of the paper. The calculation of these processes includes the application of cuts to identify photons and leptons. In order to provide an additional cross-check we have adopted the sets of cuts used in Ref. [24] for the processes in Eq. (3.6). The cuts for the processes are given in Tables 7-10 of that reference. For the convenience of the reader all the cuts that we use for the various processes are given in appendix C. We choose a common renormalization and factorization scale µ that, however, depends upon the process as follows:

Input parameters
Most parameters are specified in Table 3, where for generality we have identified values in the complex mass scheme [142]. However, for the calculation of the inclusive cross-sections (i.e. Z, W ± , H, ZH and W ± H production) described in section 3.1 all parameters are kept real by setting Γ Z = Γ W = Γ H = 0.
For W ± production we use a CKM matrix that employs the 2016 PDG values [143]: In all other processes we use a diagonal CKM matrix. We use n f = 5 flavours of massless partons throughout, except for the W − W + process that uses n f = 4 to eliminate contributions at NNLO from final states such as W − W + bb (that are considered a part of tt .56246890198475 × 10 −3 giving 1/α ≈ 132.23 . . . Table 3. Input and derived parameters used for our numerical estimates. production and subsequent decay). For the W ± H and ZH processes, diagrams in which the Higgs boson couples directly to a top quark loop are computed in the effective theory that is valid in the large m t limit. Contributions from a massive top quark are neglected in the virtual 2-loop matrix elements for all processes, and also throughout the calculation of NNLO corrections to the W − W + and Zγ processes. All calculations are performed at √ s = 13 TeV and we use the NNPDF3.0 set of parton distribution functions [144], with the set matched to the order of the calculation and the number of quark flavours.

NLO
We first provide a set of illustrative results by computing results at NLO accuracy, using both non-local slicing methods. At this order we can also compute benchmark results directly in MCFM using the subtraction method [145] in the dipole formulation [146].

Inclusive processes
The NLO results for inclusive production are shown in Fig. 1 (2 → 1 processes) and Fig. 2 (2 → 2 associated Higgs production processes). For W H production we show the result for the sum of the two W charges, σ(W ± H) = σ(W + H) + σ(W − H). The result for the NLO cross section is computed as a function of τ for 0-jettiness, c.f. Eq. (3.5), and as a function of T for q T -slicing, c.f. Eq. (3.4). Since the value of the cutoff used to present the NNLO results in Ref. [24] is T = q cut T /Q = 0.15% we perform the calculation at values of with this as a lower bound. We note that setting τ = 0.15% corresponds to τ cut /Q ≈ 10 −4 , c.f. Eq. (3.5).
The results from the non-local slicing schemes are compared to those of MCFM dipolesubtraction calculations, which also all agree fully with the results reported in Table 6 of Ref. [24]. A fit to the results at fixed values of T and τ is performed using the form, where r = 2 (q T ) and r = √ 2 (0-jettiness) effectively undoes the rescaling introduced in Eq. (3.5). This fit form anticipates the effect of possible power corrections to the factoriza-tion theorems used in obtaining the below-cut contribution (quadratic for q T and linear for 0-jettiness) but here we only use this fit to guide the eye.
In all cases the results from the non-local slicing calculations approach the known cross sections as the cutoff approaches zero. For H, W ± H and ZH production the residual difference from the known result is smaller for q T than 0-jettiness slicing, for results at equal values of T and τ . For the Z and W ± processes the relative ordering is reversed. We note that all the points in these plots have been obtained by running the MCFM code with a target Monte Carlo precision that is the same for 0-jettiness and q T slicing, so that the statistical errors on data points of equal T and τ are similar. The running time of the code to reach this level of precision is essentially the same for the two non-local slicing methods, thereby providing an indirect confirmation of the scaling behavior introduced in Eq. (3.3).

Diboson production
Corresponding results for processes involving a photon are shown in Fig. 3 and, for the remaining diboson cases, in Fig. 4. Note that here, since the definition of these processes includes the application of fiducial cuts, we have fixed r = 1 in Eq. (3.8) for q T -slicing to anticipate the presence of linear power corrections. In Fig. 3, processes in which a final-state photon is observed, the approach to the known result is almost identical for 0-jettiness and q T slicing. This is also true for the ZZ process, but for the other diboson processes q T slicing is much closer to the correct result than 0-jettiness for equal values of T and τ .
For the diphoton case we have also investigated the use of "product cuts", as advocated in Ref. [93], rather than the asymmetric cuts that are our default choice. Since we already observe no pathology in the asymmetric cut results of Fig. 3 the corresponding results for product cuts are qualitatively similar and we do not show them separately here. Since the study of Ref. [93] is motivated by sensitivity specifically arising from the 2-body decay of a parent particle this is expected; the rapidly falling p T spectrum in the continuum pp → γγ case mitigates any similar issue here.       Table 6 of Ref. [24]) and the target NNLO coefficients (δ N N LO , with The result of the MCFM calculation (0-jettiness, fit result b 0 from Eq. (3.9)) is shown in the final column.

NNLO
Having established the format and pattern of results at NLO, we now turn our attention to NNLO. At this order we may compare with the results of Ref. [24] for most processes and for the remaining W ± H and ZH processes with the code vh@nnlo [147,148]. To focus more closely on the behaviour of the calculation at this order we will show results not for the total NNLO cross section, but for the O(α 2 s ) contribution that enters at this order. In order to extract benchmark predictions for this quantity from Refs. [24,147,148] we have computed the NLO cross section for each process using NNLO PDFs (denoted σ N LO * in the table) and subtracted these values from the corresponding NNLO results. To compare with the MATRIX cross sections this method is used to obtain results for T = 0.15% and after their extrapolation procedure. The target NNLO corrections for comparison purposes are shown in Table 4, as well as the results of the NNLO calculations using MCFM.
As at NLO, for each process we have used a target numerical precision for the calculation of each process in order to compare the jettiness and q T slicing methods. The actual precisions attained and the corresponding CPU times required, for the operating points τ = 0.15% and T = 0.15%, are shown in Table 5. The time required for each slicing method to reach a similar level of precision is very close for all processes, suggesting that the scaling introduced in Eq. (3.5) remains valid at NNLO for these values of the slicing parameters. The timings for the processes pp → e −ν e γ and pp → e + ν e γ differ the most, suggesting that the presence of an identified photon in these proceses may alter the scaling somewhat (at least, under these cuts). However, the timings are still not dissimilar, especially given the  Table 5. Summary of run parameters for the NNLO calculations presented in this paper. For each process, the table indicates the relative uncertainty on the NNLO coefficient (δ N N LO ) that is computed (for the lowest values of τ and T shown in this paper), as well as the time taken (in CPU days) to perform each calculation.
computational effort that must be employed for even small gains in numerical precision at this point (∼ 0.3% on the NNLO coefficient).

Inclusive production
Results for the NNLO corrections to the inclusive calculations considered in this paper are shown in Figs. 5 and 6. As at NLO we also show a fit to the data points, but this time using a form appropriate for power corrections that could be present at NNLO, For the 2 → 1 processes shown in Fig. 5 we also indicate the MATRIX result for T = 0.15% and the extrapolated result, as given in table 4, from the same calculation. For the associated Higgs production processes we also show the vh@nnlo results in Fig. 6. The results from the two non-local subtraction schemes are in excellent agreement in the limit → 0, and in the case of the 2 → 1 processes, also match those extracted from Ref. [24]. The approach to this limit differs substantially between the two subtraction schemes; results in the q T scheme are much closer to the asymptotic value across the range while, in contrast, 0-jettiness suffers from much larger corrections at finite values of τ .

Diboson production
Results for the NNLO corrections to the diboson processes considered in this paper are shown in Figs. 7 and 8, together with the benchmark results from table 4 (extracted from Ref. [24]). For the newly-included processes in MCFM, shown in Fig. 8, we note the excellent agreement with the previous calculations reported by the MATRIX collaboration.
As at NLO, for the processes with an identified photon in the final state the approach to the asymptotic limit is similar for both q T and 0-jettiness subtraction. This indicates that power corrections to the factorization theorems underlying Eqs. (2.6) and (2.9) are affected by the requirement of photon isolation in a similar way. For the other diboson processes 0-jettiness suffers from much larger power corrections than q T subtraction.

Comparison with Ref. [1]
In Ref. [1] Heinrich et al have produced NNLO predictions for Z-boson pair production using the 0-jettiness subtraction method to isolate the doubly unresolved region. Note that in Ref. [1] the Z's are considered on-shell, and consequently there is no need to introduce the complex mass scheme. Adjusting our input parameters accordingly, we obtain the results shown in Table 6. Excellent agreement with the earlier calculation is observed.  Table 6. Comparison with the on-shell ZZ results using NNPDF3.0 from Ref. [1]. The quoted uncertainties at NNLO correspond to those obtained by scale variation according to the procedure described in this reference. .

Conclusion
Our intent in the current paper has been to increase the range of processes which are available in the MCFM package at NNLO in QCD. Specifically we have added the processes, W + W − , W ± Z and ZZ, with leptonic decays of the W and Z bosons included. As well as its extensive range of processes available at NLO using dipole subtraction, MCFM now includes a wide range of processes at NNLO. Representative results for the processes that are now included at this order have been presented in Table 4. We have implemented two different slicing methods for the calculation of the NNLO QCD corrections to processes with colour singlet final states. Both methods use global variables to isolate the region of phase space with soft and collinear emission.
The jettiness method divides the phase space on the basis of the zero-jettiness, defined in Eq. (2.5) whereas the q T method divides the phase space on the basis of the total transverse momentum of the colour singlet particles. We find that the q T -slicing method appears to be subject to smaller power corrections in most cases, although in certain cases (W ± γ, Zγ, γγ at NNLO) the size of the power corrections is similar, or even slightly smaller for jettiness. We note that all these processes involve photon isolation cuts.
Since there is a well-developed literature on the summation of logarithms of transverse momentum the q T -slicing method is easily extended to perform resummation of logarithms of q T . For the moment the application of the q T -slicing method to coloured final states at NNLO has been limited to the case of heavy-quark production [64,65,83]. The extension to processes in which a massless parton is present in the final state is not straightforward, although there are encouraging signs at NLO using a q T surrogate, based on the k T jet algorithm [149]. Despite the larger power corrections, which tend to disfavour the jettiness slicing method, we note that the jettiness subtraction method already has the proven ability to deal with coloured final states, such as W + jet, Z + jet and H + jet. Results for these processes have already been presented with MCFM.
We have provided a detailed comparison of the two non-local subtraction methods in a way that they are directly comparable. In this paper we have avoided as much as possible the imposition of cuts, because of the influence that injudicious choices of cuts can have on the power corrections. Resummation, by reducing the sensitivity to the low q T region, has the benefit that it circumvents the additional power corrections which can occur in the presence of some fiducial cuts [92,93]. The calculations we have presented provide the basis for future extensions of MCFM that include q T resummation for diboson processes, extending the work in Refs. [96,97].

A Leading log behaviour of colour singlet production cross section
For simplicity, we shall consider the simple Drell-Yan process in leading order, although the discussion will apply mutatis mutandis to all colour singlet final states. We follow closely the discussion of ref. [150]. The lowest order cross section has the form The invariant Drell-Yan cross section with the emission of one gluon of momentum k is, k and k is the gluon momentum. Taking the gluon momentum to be k = αp 1 + βp 2 + k T the invariant gluon momentum integral becomes, The parton cross section for fixed virtual photon transverse momentum q T is, To eliminate the delta functions in these expressions it is useful to take moments with respect to ρ = Q 2 /ŝ of the partonic cross section, In the limit q T → 0 (ignoring powers of α, β in the numerator), To explicitly exhibit the double logarithms we define, Integrating over q T up to q cut T and cancelling IR singularities by inclusion of the virtual diagrams gives, the order α s expansion of Eq. (3.1). This should be compared with the jettiness calculation, Integrating over τ up to τ cut and cancelling IR singularities at τ = 0 by inclusion of the virtual diagrams gives, the order α s expansion of Eq. (3.2).

B Translation of two-loop corrections to the hard function
For the W ± γ, Zγ and γγ processes presented in Refs. [90,134] the finite remainders of the two-loop matrix elements remove singular terms of the form specified by Catani in Ref. [151] but without a factor of (−µ 2 /s) 2 in the hard radiation factor H (2) ( ). The translation from this scheme to a standard MS subtraction of the singularities, to obtain the hard functions H ij introduced in section 2, has been described in Ref. [152]. The implementation of this conversion has been discussed in some detail for these processes in Refs. [30,31,35]. 4 For the diboson (W + W − , W ± Z, ZZ) processes presented in Ref. [91] the finite remainders are presented in two schemes, in which the singularities are subtracted according either to exactly Catani's scheme [151] or to a scheme that is well-suited for the original formulation of q T subtraction [153]. Starting from the latter (Ω (n),finite q T ), we convert to amplitudes that enter the hard function (Ω where the coefficients are given by, photon cuts p T,γ 1 > 40 GeV, p T,γ 2 > 25 GeV, |η γ | < 2.5 photon isolation Frixione isolation with n = 1, ε = 0.5 and δ 0 = 0.4 jet definition anti-k T algorithm with R = 0.4; p T,j > 25 GeV, |η j | < 4.5 Table 7. Fiducial cuts for the γγ process.  In these formulae the logarithm is L = log(µ 2 /s 12 ) and analytic continuation has already been performed assuming s 12 > 0. As usual C F = 4/3, C A = 3 and ζ 3 = 1.20205690 . . .. Setting L = 0 reproduces the conversion factors presented in Eq. (2.9) of Ref. [1]. After conversion in this way the amplitudes are suitable for implementation in MCFM.

C Cuts
The fiducial cuts for the γγ production process are given in Table 7, for Zγ and W ± γ production processes in Table 8, for W W and W ± Z production processes in Table 9, and for ZZ processes in Table 10. These cuts have been deliberately chosen to be the same as Ref. [24].  Table 9. Fiducial cuts for the pp → e − µ + ν µνe (W W ) and pp → eν e µ + µ − (W ± Z) processes.