aainstitutetext: Università di Milano - Bicocca, Piazza della Scienza 3, 20126
Milano, Italybbinstitutetext: INFN, Sezione di Milano-Bicocca, Piazza della
Scienza 3, I-20126 Milano, Italyccinstitutetext: INFN, Sezione di Milano &
TIFLab, Via Celoria 16, I-20133 Milano, Italy

# An interface between the Powheg Box and MadGraph5_aMC@NLO

Paolo Nason a,b Carlo Oleari a,b Marco Rocco c Marco Zaro
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we present a framework for developing Powheg Box generators
using MadGraph5_aMC@NLO for the computation of the matrix elements. Within
this framework, all the flexibility of MadGraph5_aMC@NLO for the generation of
matrix elements for Standard Model processes and for several of its extensions
can be exploited, as well as all features of the Powheg Box framework,
including the possibility of multijet merging without a merging scale (using
the so called MiNLO approach). As a proof of concept, we develop a generator
for the production of a spin-0 Higgs-like boson in association with up to two
jets, with CP-violating couplings.

###### Keywords:

NLO, parton shower, automatic generators.

TIF-UNIMI-2020-20

##  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
Frixione:1995ms ; Catani:1996vz , and, on the other, by the theoretical
developement of algorithms for interfacing fixed order calculations with
parton shower generators Frixione:2002ik ; Nason:2004rx ; Frixione:2007vw ;
Jadach:2015mza ; Alioli:2012fc ; Lonnblad:2012ix . These algorithms were
implemented in software packages for the automatic computation of NLO
corrections Alwall:2014hca ; Cascioli:2011va ; Cullen:2011ac ; Actis:2016mpe ;
Berger:2008sj , and for the automatic implementation of NLO+PS generators
Alioli:2010xd ; Alwall:2014hca ; Gleisberg:2003xi ; Bahr:2008pv ;
Alioli:2015toa  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 Degrande:2011ua
, for example as generated by FeynRules Christensen:2008py ; Alloul:2013bka .
In particular, in order to undertake a 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 Degrande:2014vpa . 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 Frixione:2019fxg , 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. Frixione:2019fxg  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.111A
variant of the MC@NLO method for drastically reducing the negative weight
fraction has appeared in Ref. Frederix:2020trv . 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 Hamilton:2012np ; Hamilton:2012rf . 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
Hamilton:2013fea ; Monni:2019whf .222Alternative methods for multijet merging
have been presented in Refs. Frederix:2012ps ; Lonnblad:2012ix ; Alioli:2012fc
. Alternative methods for NNLO+PS accuracy have been proposed in Refs.
Alioli:2013hqa ; Hoeche:2014aia . 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 Jezo:2015aia .333See also Ref.
Frederix:2016rdc .

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 Campbell:2012am ,
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. Luisoni:2013cuh ; Luisoni:2015mpa  for Gosam, and in Ref. Jezo:2016ujg
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.444Versions 2.6 and onward are fully supported, for what concerns QCD
corrections. The extension of the interface to more recent releases able to
deal with electro-weak corrections (from version 3) Frederix:2018nkq  is left
for future work. It re-organises the output of MG5_aMC in a format which is
suitable for the Powheg Box Alioli:2010xd , closely following what is
described in Ref. Campbell:2012am . 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 Hirschi:2011pa ; Alwall:2014hca , which encapsulates several
different strategies, such as integrand reduction Ossola:2006us , Laurent-
series expansion Mastrolia:2012bu  and tensor-integral reduction
Passarino:1978jh ; Davydychev:1991va ; Denner:2005nn , as implemented in
different computer libraries Ossola:2007ax ; Peraro:2014cba ; Hirschi:2016mdz
; Denner:2016kdg  and improved by an in-house implementation of the OpenLoops
method Cascioli:2011va . 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 deAquino:2011ub .

###  2.1 Technical details

The interface plugin, dubbed MG5aMC-PWG, is publicly available.555
https://code.launchpad.net/~mg5amc-pwg-team/mg5amc-pwg/v0 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,

    ./bin/mg5_aMC --mode=MG5aMC_PWG

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:

    generate p p > t t~ [QCD]
    output pp_ttx

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 Degrande:2011ua  used in
MG5_aMC.666It should be noted that the ``Cards/param_card.dat! file is not
read at execution time. Rather, it is parsed at compilation time into a
Fortran include file, which is then compiled together with the code. Hence,
after any parameter modification within this file, the main executable has to
be recompiled. 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 $X_{0}jj$, 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). 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

        (\ 0, 0, mdl_mt, mdl_mt, 0 \)

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

        ./prepare_run_dir test

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 Nason:2004rx , and
applied for the first time in Ref. Alioli:2008tz .

###  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: $\boldsymbol{X_{0}jj}$ 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. Demartin:2014fia , 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 $ggX_{0}$ vertex. The starting point
is the effective Lagrangian

${\cal
L}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}_{0}=-\bar{\psi}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}\left(k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Htt}\,g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Htt}\,\cos\alpha+i\,k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Att}\,g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Att}\,\sin\alpha\,\gamma_{5}\right)\psi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}\,X_{0}\,,$
(1)

where $X_{0}$ is the spin-0 boson,
$\psi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}$
the top-quark spinor, $\alpha$ the CP-mixing angle parameter
($0\leq\alpha\leq\pi$),
$k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Htt}$
and
$k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Att}$
the real coupling parameters and

$g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Htt}=g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Att}=\frac{m_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}}{v}=\frac{y_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}t}}{\sqrt{2}}$
(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
Artoisenet:2013puc , reads

${\cal
L}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
loop}_{0,\,g}=-\frac{1}{4}\left(k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Hgg}\,g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Hgg}\,\cos\alpha\,\,G^{a}_{\mu\nu}\,G^{a,\mu\nu}+k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Agg}\,g_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Agg}\,\sin\alpha\,\,\epsilon^{\mu\nu\rho\sigma}\,G^{a}_{\mu\nu}\,G^{a}_{\rho\sigma}\right)X_{0}\,,$
(3)

where $G^{a}_{\mu\nu}$ is the gluon field strength and

$k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Hgg}=-\frac{\alpha_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
S}}{3\pi v}\,,\qquad\qquad
k_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}Agg}=\frac{\alpha_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
S}}{2\pi v}\,.$ (4)

The theoretical setup is made available online in the FeynRules Alloul:2013bka
repository as a UFO model named HC_NLO_X0 Maltoni:2013sma ; Demartin:2014fia ;
Demartin:2015uha ; Demartin:2016axk , 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:

    import model HC_NLO_X0_UFO-heft
    generate p p > x0 j j / t [QCD]
    install ninja
    install collier
    output X0jj

where we have also inserted the command lines to install ninja Peraro:2014cba
and collier Denner:2016kdg , 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 Ball:2012cx  for the parton
distribution functions, within the LHAPDF interface Whalley:2005nh ;
Buckley:2014ana .

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

$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{k}}>10\leavevmode\nobreak\
{\rm GeV}\,,\quad k=1,2\,,\qquad\qquad
m_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}>10\leavevmode\nobreak\
{\rm GeV}\,.$ (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.777We have set bornsuppfact to 30 GeV in our simulation.

###  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
$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}$, 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

$\mu_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
R}=\mu_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
F}=\frac{H_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}}{2}\,,$ (6)

where
$H_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}$ is the sum of the transverse masses of the particles in the final state.

Jets are reconstructed employing the
anti-$k_{\rm\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}T}$
algorithm Cacciari:2008gp  via the FastJet implementation Cacciari:2011ma ,
with distance parameter $R=0.4$, and the two leading jets are required to have
transverse momentum and pseudorapidity such that

$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{k}}>30\leavevmode\nobreak\
{\rm
GeV}\,,\qquad|\eta_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{k}}|<4.5\,,\qquad
k=1,2\,.$ (7)

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

Figure 1: Differential cross section as a function of the invariant-mass
distribution of the two leading jets in $pp\rightarrow X_{0}\hskip 0.5ptjj$
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
$\cos\alpha=1/\sqrt{2}$.

In Fig. 1 we plot the differential cross section for $X_{0}jj$ 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
Hagiwara:2009wt , 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.

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 pane, a cut of 250 GeV is imposed on the dijet mass, while on the
right pane a cut of 500 GeV is applied. The colour code is the same as in Fig.
1.

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 pane, a cut of 250 GeV is imposed on the dijet mass, while on the right
pane a cut of 500 GeV is applied. The colour code is the same as in Fig. 1.

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 pane, a cut of 250 GeV is imposed on the dijet mass, while on the right
pane a cut of 500 GeV is applied. The colour code is the same as in Fig. 1.

Figure 5: Normalised differential cross section as a function of the
pseudorapidity of the leading jet, for the three CP scenarios. On the left
pane, a cut of 250 GeV is imposed on the dijet mass, while on the right pane a
cut of 500 GeV is applied. The colour code is the same as in Fig. 1.

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

$m_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}>250\,\,{\rm
GeV}\qquad{\rm and}\qquad
m_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}>500\,\,{\rm
GeV}\,.$ (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
$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}$ 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
$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}$
and
$\eta_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}$,
also in agreement with what is found in Ref. Demartin:2014fia .888A possible
concern is to what extent the effective-field-theory (EFT) Lagrangian of Eq.
(3) produces sound results in the high-energy regimes, since it describes the
full theory in the heavy-top-quark limit. From the exact calculation of Ref.
DelDuca:2001fn , it is known that the EFT closely reproduces the
$m_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}$
spectrum even in the very high invariant-mass region. However, the EFT
approximation breaks down when the transverse momenta of the jets are larger
than the top mass DelDuca:2001eu , overestimating the exact prediction when
$p_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}$
is larger than the top mass. Since the region of interest for discriminating
the CP properties is at low transverse momentum, we can trust the results
obtained within the EFT approach.

Figure 6: Normalised differential cross section as a function of the
pseudorapidity separation of the two leading jets (see eq. (9)), for the three
CP scenarios. On the left pane, a cut of 250 GeV is imposed on the dijet mass,
while on the right pane a cut of 500 GeV is applied. The colour code is the
same as in Fig. 1.

Figure 7: Normalised differential cross section as a function of the azimuthal
separation of the two leading jets (see eq. (10)), for the three CP scenarios.
On the left pane, a cut of 250 GeV is imposed on the dijet mass, while on the
right pane a cut of 500 GeV is applied. The colour code is the same as in Fig.
1.

The most sensitive observables to the CP coupling of the $X_{0}$ boson to the
top quark in gluon fusion are dijet-correlation variables Plehn:2001nj ;
Klamke:2007cu ; Hagiwara:2009wt ; Andersen:2010zx ; Campanario:2010mi ;
Englert:2012ct ; Englert:2012xt ; Dolan:2014upa . 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

$\Delta\eta_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}=\left|\eta_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}-\eta_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right|.$
(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
Hagiwara:2009wt . In fact, the shape of the differential cross sections as a
function of
$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}$
are very different, as shown in Fig. 7, where we have defined (modulo $2\pi$)

$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}=\left|\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}-\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right|,$
(10)

where the azimuth of a jet is computed as

$\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{k}}=\arg\left({\bf
p}^{j_{k}}\cdot\hat{y}+i\,{\bf p}^{j_{k}}\cdot\hat{x}\right),\quad k=1,2\,,$
(11)

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

Figure 8: Normalised differential cross section as a function of the oriented
azimuthal separation of the two leading jets, defined in Eq. (12), for the
three CP scenarios. On the left pane, a cut of 250 GeV is imposed on the dijet
mass, while on the right pane a cut of 500 GeV is applied. The colour code is
the same as in Fig. 1.

As pointed out in Refs. Klamke:2007cu ; Hankele:2006ma , 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. Gritsan:2020pib
, namely

$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
or}\equiv\frac{\left(\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}\times\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right)\cdot\hat{z}}{\left|\left(\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}\times\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right)\cdot\hat{z}\right|}\,\frac{\left({\bf
p}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}-{\bf
p}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right)\cdot\hat{z}}{\left|\left({\bf
p}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}-{\bf
p}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right)\cdot\hat{z}\right|}\,\arccos\\!{\left(\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}}\cdot\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{2}}\right)}\,,$
(12)

where $\hat{\bf
p}_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
T}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{k}}$
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
$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
or}$, are shown in Fig. 8, and their shape is visibly different.

Figure 9: Normalised differential cross section as a function of the oriented
azimuthal separation of the two leading jets, defined in Eq. (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.

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
$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}$cannot
distinguish between them.

###  3.5 Reweighting

Figure 10: Normalised differential cross section as a function of the oriented
azimuthal separation of the two leading jets, defined in Eq. (12), with a cut
of 250 GeV imposed on the dijet mass. On the left pane, 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 pane, 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.

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).
Figure 12: Same as Fig. 10 but for the reweighting of the scalar sample to the
CP scenario defined by $\cos\alpha=0.985$.

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
$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
or}$ distribution obtained from the original scalar sample is plotted in
dotted blue, on both panes. This curve corresponds to the $0^{+}$ line on the
left pane of Fig. 8. The scalar sample is reweighted to the pseudoscalar
scenario on the left pane and to the mixed scenario on the right pane. 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 $+10\%$.

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
$\Delta\phi_{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}j_{1}j_{2}}^{\mathchoice{\displaystyle}{\scriptstyle}{\scriptscriptstyle}{\scriptscriptstyle}\rm
or}$ 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 pane, 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. Hamilton:2012rf ; Frederix:2015fyz .

Figure 13: On the left pane 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 pane in red, the inclusive
rapidity of the $X_{0}$ boson, for the same CP scenario as in the left pane.
Both plots are obtained with MiNLO.

##  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 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.

## Acknowledgments

P.N. acknowledges support from Fondazione Cariplo and Regione Lombardia, grant
2017-2070, and from INFN. We thank S. Frixione and F. Maltoni for useful
discussions. We thank A. Gritsan for suggesting the process we implemented.

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