1 Introduction

Next-to-leading-order (NLO) calculations for Standard Model (SM) and, sometimes, beyond-the-SM (BSM) processes, interfaced to parton shower (PS) generators, generally dubbed NLO+PS generators, are by now the methods of choice for the generation of event samples for signal and background processes at the LHC. This state of the art has been made possible, on the one side, by the formulations of general methods for computing NLO corrections [1, 2], and, on the other, by the theoretical development of algorithms for interfacing fixed order calculations with parton shower generators [3,4,5,6,7,8]. These algorithms were implemented in software packages for the automatic computation of NLO corrections [9,10,11,12,13], and for the automatic implementation of NLO+PS generators [9, 14,15,16,17] that considerably ease the construction of generators for new processes.

MadGraph5_aMC@NLO, often abbreviated to MG5_aMC in the following, is a framework where automation has been pushed to the highest level. In fact, a user without any knowledge of NLO calculations or NLO+PS implementations can easily generate samples of parton-level events with NLO+PS accuracy, within the MC@NLO procedure. These events can be then directly fed into a PS generator, such as Pythia or Herwig. The MG5_aMC framework is not restricted to the case of SM processes. In fact, it is possible to employ any user-defined model if this is provided in the so-called UFO format [18], for example as generated by FeynRules [19, 20]. In particular, in order to undertake an NLO computation, the model should include the relevant UV and rational counterterms (both needed for the numerical evaluation of the one-loop matrix elements), which can be also automatically computed with FeynRules+NLOCT [21]. Furthermore, the FeynRules+MG5_aMC framework has been recently extended in order to fully support the supersymmetric case, including the implementation of different renormalisation conditions [22], and the use of the so-called diagram removal and diagram subtraction techniques when intermediate resonances are present. NLO capabilities for BSM processes have been proven successful for a number of processes, see Ref. [22] and references therein.

The Powheg method allows to generate events with positive weights and, because of this, it has become the method of choice when large samples of events are needed. In fact, in view of the large amount of computer resources needed for detector simulation, the experimental collaborations cannot afford to use the larger samples that are required when negative weights are present.Footnote 1 The method has been also extended with the introduction of some theoretical developments of general interest. One of them deals with the generation of multijet samples that maintain a certain level of accuracy, even when some of the jets become unresolved [24, 25]. This approach has also led to the development of NNLO+PS generators, i.e. generators where next-to-next-to-leading-order (NNLO) calculations are interfaced to parton showers [26,27,28].Footnote 2 Another development has been the extension of the Powheg method for the inclusion of processes with decaying coloured resonances, which is capable of handling the interference of the emitted radiation generated in production and decay [32].Footnote 3

The Powheg Box framework automatises the construction of NLO+PS generators, once the matrix elements are available. In the early Powheg Box processes, the matrix elements were obtained from the authors of specific calculations. A considerable leap in the construction of the matrix elements took place when an interface of the Powheg Box to MadGraph4 was set up [34], allowing for the implementation of all tree-level ingredients required by a given NLO process. After this development, the only missing ingredient for an NLO calculation in the Powheg Box was the virtual contribution. Later, interfaces to automatic generators of virtual processes were also developed in Refs. [35, 36] for Gosam, and in Ref. [37] for OpenLoops.

As of now, an interface to the matrix-element generator that is available within the MG5_aMC package has not been developed. The main obstacle is the fact that MG5_aMC is built as a single package that aims at the production of partonic events, at difference with MadGraph4, that was initially conceived for the generation of tree-level matrix elements. An interface between the matrix-element generator of MG5_aMC and the Powheg Box is also highly desirable since many BSM processes are available within MG5_aMC. In order to exploit the full capabilities of the MG5_aMC package, such interface should also build, in addition to the virtual contribution, all the necessary tree-level matrix elements: the Born, the colour- and spin-correlated Born, and the real matrix elements.

The purpose of the present work is to present an interface between the MG5_aMC matrix-element generator and the Powheg Box. The structure of the interface is such that developments in MG5_aMC and Powheg can remain independent to a large extent. For this reason, our aim is not to construct a framework that is automatised at the same level as the full MG5_aMC package itself, but rather to build an MG5_aMC extension that makes the NLO matrix elements readily available to Powheg. Thus, progresses on the Powheg Box side and on the MG5_aMC side can take place independently, which is a considerable advantage in view of the way in which theoretical projects are developed. Furthermore, this kind of interface allows generalisations to other NLO+PS frameworks, that may also benefit from it for the implementation of the matrix elements.

The paper is organised as follows. In Sect. 2 we describe the interface and we give some technical details on how to use it and how to distribute the generated code. In Sect. 3 we consider, as a case study, the production of a spin-0 boson \(X_0\) plus two jets. In particular, we present a few distributions able to characterise the \(X_0\) boson CP properties and we discuss some features connected to the Powheg Box reweighting feature. We also show a few distributions obtained with the MiNLO approach. Finally, in Sect. 4 we draw our conclusions.

2 Interface to MG5_aMC

The new interface between Powheg and MG5_aMC uses the capability of the latter to provide tree-level and one-loop matrix elements to be used by the former. The interface itself is a plugin for MG5_aMC: as such, it does not require any modification of the core code and it works with any recent version of MG5_aMC.Footnote 4 It re-organises the output of MG5_aMC in a format which is suitable for the Powheg Box [14], closely following what is described in Ref. [34]. At variance with what is discussed there, no external providers for the one-loop matrix elements are needed. Rather, one-loop matrix elements are directly generated by MG5_aMC thanks to the MadLoop module [9, 39], which encapsulates several different strategies, such as integrand reduction [40], Laurent-series expansion [41] and tensor-integral reduction [42,43,44], as implemented in different computer libraries [45,46,47,48] and improved by an in-house implementation of the OpenLoops method [10]. Thus, by fully exploiting the capabilities of MadLoop, the evaluation of virtual matrix elements and the assessment of the numerical stability of the results are granted. Along with the matrix elements, the relevant helicity routines are also provided, in the ALOHA format [49].

2.1 Technical details

The interface plugin, dubbed MG5aMC-PWG, is publicly available.Footnote 5 Its usage is very simple, as one only needs to copy (or link) the MG5aMC_PWG folder inside the PLUGIN directory of MG5_aMC. Please refer to the README file enclosed in the package for conditions of usage and instructions.

The plugin can be loaded by launching, within the MG5_aMC installation directory,

figure a

in a command shell. In order to generate the code for a specific process at NLO QCD accuracy, the usual syntax of MG5_aMC should be employed. For example, in the case of top-pair production, the syntax is the following:

figure b

where pp_ttx is the name (chosen by the user) of the directory where the code will be created. During the execution of the generate command, the MG5aMC-PWG plugin checks whether an installation of the Powheg Box V2 is available on the system and asks for its installation path (this is needed only once).

When this stage is concluded, the user can quit MG5_aMC and finds the MG5_aMC code for the Born, real and virtual contributions in the pp_ttx directory, in addition to a few basic Powheg Box V2 files. In particular, the Born.f, real.f and virtual.f files are ready to be used. Also the init_processes.f file can be used as it is, but can be also modified if particular features of the Powheg Box V2 need to be activated and initialised.

A few comments about the other files are in order:

  • The Born_phsp.f file is just a place holder. It needs to be replaced by the actual phase-space generator for the process at hand. Examples of Born_phsp.f implementations can be found in the processes already implemented in the Powheg Box V2. In the current setup, a subroutine born_suppression should be also implemented in the Born_phsp.f file. This function is used at the integration stage to suppress divergences when present at the Born level, i.e. when there are jets and photons.

  • The call of the setpara(“param_card.dat”) routine in the init_couplings.f file initialises the parameters listed in the Cards/param_card.dat file to the corresponding values, according to the UFO model [18] used in MG5_aMC.Footnote 6 It is also possible to assign a value to a MG5_aMC parameter at execution time. An example of this can be found in the init_couplings.f file for the process \(X_0jj\), that we discuss in Sect. 3. In this file we reassign the value of \(\cos \alpha \), the CP-mixing parameter that appears in the Lagrangian of Eq. (3.3). This parameter is indicated with cosa in the Cards/param_card.dat file, and is initialised to the value specified in this file, if no further action is taken. In order to reassign its value at execution time, we change the values of the internal MG5_aMC variables, mdl_cosa and mp_mdl_cosa (for double and quadruple precision), that encode this parameter.

    After any reassignment of the MG5_aMC parameters, the user has to call the coup routine in order to recompute all the dependent variables.

  • In order to have full consistency between the MG5_aMC amplitudes and what is computed by the Powheg Box V2, all the physical parameters used by the Powheg Box V2 should be set starting from those assigned or computed by MG5_aMC. An example of this is the list of the external-particle masses, kn_masses, used by Powheg Box V2 when generating the kinematics of the event. Using \(t \bar{t} \) production as example, kn_masses should be set to

    figure c

    in init_couplings.f or Born_phsp.f, where mdl_mt is the mass of the top quark used inside MG5_aMC, the first two entries are the masses of the incoming particles, and the last massless particle is the radiated one, when computing the real contribution.

  • The interface also builds a script file, prepare_run_dir, that is useful to create a directory where the produced code can be executed. For example, by typing the command

    figure d

    a directory test is created. This directory contains all the relevant links to the MG5_aMC code and a template of the powheg.input file, required by the Powheg Box V2. This last file should then be changed and modified according to the process at hand.

The Powheg process generated along these lines can be completed with all sorts of features that are commonly used in the Powheg Box V2. For example, one can activate the MiNLO option for processes with associated jets, or use the damping option to separate the real contributions into two parts, along the lines of what was suggested in the original Powheg paper [4], and applied for the first time in Ref. [50].

2.2 Distribution of the code

A process generated with this interface to MG5_aMC cannot be distributed as a usual Powheg Box process, since the searching path of the linked libraries are written in several files at generation time.

An author can distribute the instructions for MG5_aMC, needed in order to generate the process, and the actual files, that overwrite the place holders created by the interface plugin. In this way, all relevant paths point to the right directories in the user computer.

Alternatively, the author of the process may provide a script file that automatically executes all these tasks, helping the installation phase.

3 A case study: \({X_0jj}\) production with CP-violating couplings

For our case study, we considered the production of a spin-0 boson \(X_0\) (a Higgs-like boson) that couples to a massive top quark, produced via gluon fusion, and accompanied by two jets, in the heavy-top-mass limit. We discuss a few distributions able to characterise the \(X_0\) boson CP properties, and discuss a few results obtained using the Powheg Box V2 reweighting feature. We also present a few distributions obtained with the MiNLO method.

3.1 Theoretical setup

The theoretical framework of this study is fully inherited from what was done in Ref. [51], where the process was studied at NLO in QCD. In particular, in the heavy-top-mass limit, the CP structure of the \(X_0\)-top interaction characterises the effective \(gg X_0\) vertex. The starting point is the effective Lagrangian

$$\begin{aligned} \mathcal{L}^{t}_0= -\bar{\psi }_{t}\left( k_{Htt}\,g_{Htt}\,\cos \alpha + i\,k_{Att}\,g_{Att}\,\sin \alpha \,\gamma _5 \right) \psi _{t}\,X_0, \end{aligned}$$
(3.1)

where \(X_0\) is the spin-0 boson, \(\psi _{t}\) the top-quark spinor, \(\alpha \) the CP-mixing angle parameter (\(0 \le \alpha \le \pi \)), \(k_{Htt}\) and \(k_{Att}\) the real coupling parameters and

$$\begin{aligned} g_{Htt}= g_{Att}= \frac{m_{t}}{v} = \frac{y_{t}}{\sqrt{2}} \end{aligned}$$
(3.2)

the Yukawa couplings, with v the vacuum expectation value.

The CP-even case, that will be labeled \(0^+\), corresponds to the assignment \(\cos \alpha = 1\), namely to the SM scenario, while the CP-odd case, labeled \(0^-\), to \(\cos \alpha = 0\). A CP-mixed case, \(0^\pm \), where the spin-0 boson receives contributions from both a scalar and a pseudoscalar state, is also taken into account by setting \(\cos \alpha = 1/\sqrt{2}\).

For our purposes, it will suffice to notice that the Higgs interaction with the gluons originates as an effective coupling induced by a top-quark loop. The relevant effective Lagrangian, in the Higgs Characterisation framework [52], reads

$$\begin{aligned} \mathcal{L}^{\mathrm{loop}}_{0,\,g}= & {} -\frac{1}{4} \left( k_{Hgg}\,g_{Hgg}\,\cos \alpha \,\,G^a_{\mu \nu }\,G^{a,\mu \nu }\right. \nonumber \\&\left. +k_{Agg}\,g_{Agg}\,\sin \alpha \,\, \epsilon ^{\mu \nu \rho \sigma }\,G^a_{\mu \nu }\,G^a_{\rho \sigma } \right) X_0, \end{aligned}$$
(3.3)

where \(G^a_{\mu \nu }\) is the gluon field strength and

$$\begin{aligned} k_{Hgg}= -\frac{\alpha _\mathrm{S}}{3\pi v}, \qquad \qquad k_{Agg}= \frac{\alpha _\mathrm{S}}{2\pi v}. \end{aligned}$$
(3.4)

The theoretical setup is made available online in the FeynRules [20] repository as a UFO model named HC_NLO_X0 [51, 53,54,55], which is in fact the one used for our case study.

3.2 Generation of the code

In order to generate the code, we have first to install the UFO model HC_NLO_X0_UFO.zip under the models directory of the MG5_aMC version being used. We have then followed the procedure described in Sect. 2.1 for the generation of the code, and given the following commands to MG5_aMC:

figure e

where we have also inserted the command lines to install ninja [46] and collier [48], that are optional and need to be installed just once.

We have then overwritten the Born_phsp.f file generated by the interface with the Born_phsp.f from the Hjj Powheg Box V2 process, taking care of assigning to the Powheg variables hmass and hwidth (the mass and width of the Higgs-like boson) the MG5_aMC values, mdl_mx0 and mdl_wx0 respectively.

In order to ease the installation procedure, we provide a tarball file that needs to be inflated in the installation directory. This file contains all the modified files that replace the place holders.

3.3 Simulation parameters

We have performed a simulation for the LHC, running at a centre-of-mass energy of \(\sqrt{S}=13\) TeV. The mass of the spin-0 boson \(X_0\) has been set equal to 125 GeV. We have chosen the NNPDF2.3 (NLO) set [56] for the parton distribution functions, within the LHAPDF interface [57, 58].

The differential cross section for \(X_0 jj\) production is already divergent at the Born level, unless a minimum set of generation cuts is imposed on the transverse momentum of the final-state jets and on their invariant mass. Alternatively, the divergences can be avoided if the code is executed with the MiNLO option activated. We have generated the kinematics of the underlying Born configurations imposing the following minimum set of cuts

$$\begin{aligned} p_\mathrm{T}^{j_k}>10~\mathrm{GeV}, \quad k = 1,2, \qquad \qquad m_{j_1 j_2}> 10~\mathrm{GeV}. \end{aligned}$$
(3.5)

In the phenomenological study we perform in Sect. 3.4, we apply more stringent cuts, and we have checked that the results we present are insensitive to the generation cuts.

In order to integrate the divergent underlying Born cross section, the Powheg Box V2 can further apply a suppression factor at the integrand level. We stress that the final kinematic distributions are independent of this factor.Footnote 7

3.4 Phenomenology

In this section we present results produced by the Powheg Box V2 at the Les Houches Event (LHE) level, i.e. after the emission of the first radiation, accurate at NLO for large transverse momentum, and with leading-logarithmic accuracy at small \(p_{\mathrm{T}}\), due to the presence of the Powheg Sudakov form factor. The results are computed on samples of 3.2 M events.

The renormalisation and factorisation scales are set to

$$\begin{aligned} \mu _\mathrm{R} = \mu _\mathrm{F} = \frac{H_\mathrm{T}}{2}, \end{aligned}$$
(3.6)

where \(H_\mathrm{T}\) is the sum of the transverse masses of the particles in the final state.

Jets are reconstructed employing the anti-\(k_{\mathrm{T}} \) algorithm [59] via the FastJet implementation [60], with distance parameter \(R=0.4\), and the two leading jets are required to have transverse momentum and pseudorapidity such that

$$\begin{aligned} p_\mathrm{T}^{j_k} >30~\mathrm{GeV}, \qquad |\eta _{j_k}| < 4.5,\qquad k = 1,2. \end{aligned}$$
(3.7)

Events that do not pass this minimum set of acceptance cuts are discarded.

Fig. 1
figure 1

Differential cross section as a function of the invariant-mass distribution of the two leading jets in \(pp\rightarrow X_0jj\) for the three CP scenarios. The blue curve corresponds to the CP-even scenario with \(\cos \alpha = 1\), the red curve to the CP-odd scenario with \(\cos \alpha = 0\) and the black curve to the mixture of the \(0^+\) and \(0^-\) scenarios with \(\cos \alpha = 1/\sqrt{2}\)

Full size image

In Fig. 1 we plot the differential cross section for \(X_0jj\) production as a function of the invariant mass of the two leading jets, \(m_{j_1 j_2}\), for three different CP scenarios: CP even (\(0^+\)), CP odd (\(0^-\)) and a mixture of the two (\(0^\pm \)). The shapes of the three spectra are very similar among each other. Since a cut on the invariant mass of the dijet system enhances the discriminating power among different CP scenarios [61], the fact that the three spectra have similar shapes implies that the cut acts in a similar way on each of them. Typically a cut on \(m_{j_1 j_2}\) enhances the contributions coming from the exchange of a gluon in the t channel, and these contributions are more sensitive to the CP properties of the \(X_0\) boson.

Fig. 2
figure 2

Normalised differential cross section as a function of the transverse momentum of the spin-0 boson \(X_0\), for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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Fig. 3
figure 3

Normalised differential cross section as a function of the pseudorapidity of the spin-0 boson \(X_0\), for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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Fig. 4
figure 4

Normalised differential cross section as a function of the transverse momentum of the leading jet, for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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Fig. 5
figure 5

Normalised differential cross section as a function of the pseudorapidity of the leading jet, for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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In the following plots we impose an additional cut on the dijet mass. In particular, we consider the two cases where

$$\begin{aligned} m_{j_1 j_2}> 250 \,\,\mathrm{GeV} \quad \mathrm{and} \quad m_{j_1 j_2}>500 \,\, \mathrm{GeV}. \end{aligned}$$
(3.8)

In addition, since we are interested in shape comparisons among different CP scenarios, we normalise each curve to one.

In Figs. 2 and 3 we plot the transverse momentum and pseudorapidity of the \(X_0\) boson, and in Figs. 4 and 5 we show the transverse momentum and pseudorapidity of the leading jet. The increase of the cut on the dijet mass hardens the \(p_{\mathrm{T}}\) spectrum of the \(X_0\) boson and the leading jet \(j_1\). Moreover, there are only mild differences among the three CP scenarios in the \(X_0\) distributions at low transverse momentum and in the central pseudorapidity region, with a modest enhancement when the dijet-mass cut increases. No substantial differences are present in \(p_\mathrm{T}^{j_1}\) and \(\eta _{j_1}\), also in agreement with what is found in Ref. [51].Footnote 8

Fig. 6
figure 6

Normalised differential cross section as a function of the pseudorapidity separation of the two leading jets (see Eq. (3.9)), for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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The most sensitive observables to the CP coupling of the \(X_0\) boson to the top quark in gluon fusion are dijet-correlation variables [61, 64,65,66,67,68,69,70]. As displayed in Fig. 6, no significant differences are seen in the differential cross sections as a function of the pseudorapidity separation of the two leading jets

$$\begin{aligned} \Delta \eta _{j_1 j_2}= \left| \eta _{j_1} - \eta _{j_2}\right| . \end{aligned}$$
(3.9)

Instead, when the differential cross sections are expressed as a function of the azimuthal-angle separation, the CP nature of the coupling is more evident [61]. In fact, the shape of the differential cross sections as a function of \(\Delta \phi _{j_1 j_2}\) are very different, as shown in Fig. 7, where we have defined (modulo \(2\pi \))

$$\begin{aligned} \Delta \phi _{j_1 j_2}= \left| \phi _{j_1} - \phi _{j_2}\right| , \end{aligned}$$
(3.10)

where the azimuth of a jet is computed as

$$\begin{aligned} \phi _{j_k} = \arg \left( \mathbf{p}^{j_k} \cdot \hat{y} + i\, \mathbf{p}^{j_k} \cdot \hat{x} \right) , \quad k=1,2, \end{aligned}$$
(3.11)

with \(\mathbf{p}^{j_k}\) the tri-momentum of the jet k and \(\hat{x}\,(\hat{y})\) the unit vector along the \(x\,(y\))-axis direction.

Fig. 7
figure 7

Normalised differential cross section as a function of the azimuthal separation of the two leading jets (see Eq. (3.10)), for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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Fig. 8
figure 8

Normalised differential cross section as a function of the oriented azimuthal separation of the two leading jets, defined in Eq. (3.12), for the three CP scenarios. On the left panel, a cut of 250 GeV is imposed on the dijet mass, while on the right panel a cut of 500 GeV is applied. The colour code is the same as in Fig. 1

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As pointed out in Refs. [65, 71], a more CP-sensitive observable (especially for the maximal mixing scenario of \(\cos \alpha =1/\sqrt{2}\) considered here) is the oriented azimuthal separation of the two hardest jets. This variable contains information not only on the azimuthal separation of the two jets but also on the sign of the azimuthal angle. We have adopted the definition of this variable of Ref. [72], namely

$$\begin{aligned} \Delta \phi _{j_1 j_2}^\mathrm{or} \equiv \frac{\left( \hat{\mathbf{p}}_\mathrm{T}^{j_1} \times \hat{\mathbf{p}}_\mathrm{T}^{j_2}\right) \cdot \hat{z}}{\left| \left( \hat{\mathbf{p}}_\mathrm{T}^{j_1} \times \hat{\mathbf{p}}_\mathrm{T}^{j_2}\right) \cdot \hat{z} \right| } \, \frac{\left( \mathbf{p}^{j_1} - \mathbf{p}^{j_2}\right) \cdot \hat{z}}{\left| \left( \mathbf{p}^{j_1} - \mathbf{p}^{j_2}\right) \cdot \hat{z} \right| } \,\arccos \!{\left( \hat{\mathbf{p}}_\mathrm{T}^{j_1} \cdot \hat{\mathbf{p}}_\mathrm{T}^{j_2}\right) }, \end{aligned}$$
(3.12)

where \(\hat{\mathbf{p}}_\mathrm{T}^{j_k}\) is the jet transverse momentum, normalised to one, and \(\hat{z}\) is the unit vector along the z-axis direction.

The differential cross sections for the three different CP scenarios considered in this paper, as a function of \(\Delta \phi _{j_1 j_2}^\mathrm{or}\), are shown in Fig. 8, and their shapes are visibly different.

Fig. 9
figure 9

Normalised differential cross section as a function of the oriented azimuthal separation of the two leading jets, defined in Eq. (3.12), for the two mixed CP scenarios with \(\cos \alpha =1/\sqrt{2}\) (black curve) and \(\cos \alpha =-1/\sqrt{2}\) (grey curve). A cut of 250 GeV is imposed on the dijet mass

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In particular, the oriented azimuthal separation can also distinguish between the two scenarios with \(\cos \alpha =1/\sqrt{2}\) and \(\cos \alpha =-1/\sqrt{2}\), as illustrated in Fig. 9, while \(\Delta \phi _{j_1 j_2}\)cannot distinguish between them.

Fig. 10
figure 10

Normalised differential cross section as a function of the oriented azimuthal separation of the two leading jets, defined in Eq. (3.12), with a cut of 250 GeV imposed on the dijet mass. On the left panel, the pseudoscalar original distribution in red, the pseudoscalar as obtained by reweighting (rw) in pink, and the scalar one in dotted blue. On the right panel, the CP mixed original distribution in black, the mixed as obtained by reweighting (rw) in gray, and the scalar one in dotted blue. The ratios between the distributions obtained by reweighting and the original ones are also shown

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Fig. 11
figure 11

Same as Fig. 10 but for the reweighting of the CP mixed sample to the scalar case (on the left) and to the pseudoscalar one (on the right)

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3.5 Reweighting

In this section we present a few results obtained with the Powheg Box V2 reweighting feature. We have reweighted two of the event samples that we have produced: the scalar and the mixed one. We have then compared the reweighted distributions with the original ones, i.e. those computed from the beginning with a given value of \(\cos \alpha \). In particular, we have reweighted the scalar sample to the pseudoscalar and CP mixed cases, and we have reweighted the mixed sample to the scalar and pseudoscalar ones. We have found an overall good agreement between the reweighted and the original distributions, except for the distribution of the differential cross section expressed as a function of the oriented azimuthal angle, i.e. the distributions most sensitive to the value of the CP parameter \(\cos \alpha \).

In Fig. 10 we compare three curves. The \(\Delta \phi _{j_1 j_2}^\mathrm{or}\) distribution obtained from the original scalar sample is plotted in dotted blue, on both panels. This curve corresponds to the \(0^+\) line on the left panel of Fig. 8. The scalar sample is reweighted to the pseudoscalar scenario on the left panel and to the mixed scenario on the right panel. The reweighted sample, indicated with “rw” in the figures, is then compared with the original distribution. The ratio of the last two curves is also plotted. In both cases, in correspondence to the minima of the \(0^+\) distribution, the discrepancy between the reweighted distribution and the original one is more than \(-10\%\), the minus sign to indicate that the distributions obtained by reweighting underestimate the original ones. The opposite is also true: when the \(0^+\) distribution has maxima that are not close to the maxima of the \(0^-\) and \(0^\pm \) distributions, we have a discrepancy on the opposite side, up to \(+10\%\).

Fig. 12
figure 12

Same as Fig. 10 but for the reweighting of the scalar sample to the CP scenario defined by \(\cos \alpha =0.985\)

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Fig. 13
figure 13

On the left panel the inclusive differential cross section as a function of the transverse momentum of the hardest jet, in blue, and of the second-to-hardest one, in red. The CP scenario is defined by \(\cos \alpha =0\), namely, the pseudoscalar case. On the right panel in red, the inclusive rapidity of the \(X_0\) boson, for the same CP scenario as in the left panel. Both plots are obtained with MiNLO

Full size image

Similar conclusions can be drawn by reweighting the \(0^\pm \) sample, as illustrated in Fig. 11, in order to produce the differential cross section as a function of \(\Delta \phi _{j_1 j_2}^\mathrm{or}\) for the \(0^+\) and \(0^-\) scenarios.

These differences can be explained by noticing that the minima of the above distributions are actually zeros at LO, and the production of events around these regions is then suppressed. The reweighting procedure is not able to generate the correct distributions, if the starting one is very different from the final one, i.e., for example, going from \(\alpha =0\) to \(\alpha =\pi /2\), for the reweighting of the scalar case to the pseudoscalar one.

Otherwise, if the reweighting procedure is used to reweight distributions with similar values of the angle \(\alpha \), the procedure correctly works. This is shown in Fig. 12, where the distribution computed with \(\alpha =0\) is reweighted to the distribution with \(\alpha \sim 10^\circ \sim \pi /18\), and the agreement with the exact one is very good.

3.6 MiNLO

In this section we present a few results for the pseudoscalar \(X_0\) production, obtained within the MiNLO procedure. Although all the cuts applied on the jets in the previous sections are completely removed, the differential cross sections for inclusive quantities are finite, due to the presence of the MiNLO Sudakov form factor.

This is shown, for example, in Fig. 13, where we plot the inclusive differential cross section as a function of the transverse momentum of the hardest and of the second-to-hardest jet, on the left panel, and the inclusive rapidity of the \(X_0\) boson, on the right one.

Although finite, we cannot make any claim on the accuracy of these distributions, i.e. they do not reach the NLO accuracy of the MiNLO’ method, described in Refs. [25, 73].Footnote 9

4 Conclusions

In this paper we have presented an interface between MadGraph5_aMC@NLO and the Powheg Box V2, able to build a NLO + parton shower generator for Standard Model and many beyond-the-Standard-Model processes, in an automatic way.

The structure of the interface is such that future developments in MadGraph5_aMC@NLO and Powheg Box V2 remain independent to a large extent, so that it benefits from all the progresses coming from both sides. In fact, on the one side, MadGraph5_aMC@NLO provides the matrix elements for the Born, the colour- and spin-correlated Born, the real and the virtual contributions. On the other, the Powheg Box uses these ingredients to generate events accurate at the NLO + parton shower level. In addition, the interface writes other files needed by the Powheg Box V2. Some of them, as the list of processes, are fully finalised. Others, such as the phase-space generator, need to be adjusted in order to deal with the process at hand.

By now the interface only deals with processes for which we aim at NLO QCD accuracy. The extension including the electroweak corrections and the interface with the more recent version of the Powheg Box, i.e. the Powheg Box Res, is left as future work.

As a case study, using this interface we have generated the code for the production of a spin-0 boson plus two jets, and we have computed a few kinematic distributions, sensitive to the CP properties of the coupling of the boson with a massive top quark. We have compared these distributions with known results in the literature and found full agreement. We have also presented a few results for the pseudoscalar case, obtained within the MiNLO approach.

Finally, we have tested the Powheg Box reweighting feature. This procedure works fine for every kinematic distributions we have examined, but for the ones most sensitive to the CP nature of the \(X_0\) boson. In fact, we have observed that it works if the reweighting is done from one distribution to another, with values of the mixing angle \(\alpha \) not very different from each other.