text:
\section{Introduction}
\label{sec:intro}

The production of two isolated photons at hadron colliders, henceforth
denoted diphoton production and abbreviated as $pp\rightarrow\gamma\gamma$, is one
among the most relevant Standard Model~(SM) processes, due, on one side, to
the high production rate, and, on the other, to the relative cleanliness
of the experimental final state.
Furthermore, diphoton production is the dominant background for studies
involving Higgs boson production decaying into a photon pair, and, notably,
it was one of the dominant backgrounds for the Higgs boson
discovery~\cite{ATLAS:2012yve, CMS:2012qbp} at the Large Hadron
Collider~(LHC).  In addition, it is a background for searches for heavy
neutral resonances that can arise in a variety of beyond the Standard Model
scenarios (see e.g.~\cite{Mrenna:2000qh,Han:1998sg, Giudice:1998bp,
  Giudice:2017fmj}), and for searches for extra dimensions or supersymmetry,
and a possible channel for their detection~\cite{ATLAS:2017ayi, CMS:2018dqv}.

Within the SM, photon pairs can be produced by means of several mechanisms,
that make this otherwise simple process particularly challenging to study. In
this work we restrict to prompt photon production,\footnote{One refers to
non-prompt photons to denote photons produced from hadron decays.} where
photons are produced in the hard process. Within this category, one can
further distinguish two components: one arising from direct photon
production and the other from single and double fragmentation. In the former
case, the photons are directly produced in the hard scattering, whereas, in
the latter case, one or both photons are produced through the fragmentation
of jets. The latter mechanism can be suppressed by imposing the isolation of
the photons from the hadronic activity through a fixed or a smooth
cone algorithm~\cite{Frixione:1998jh}.
In spite of the fact that the fixed-cone
isolation criterion is simpler to apply (also at the experimental level),
only the smooth-cone isolation is suitable for theoretical calculations, as
it allows to completely remove the fragmentation component in an
infrared-safe way.

Theoretical predictions for diphoton direct production at
next-to-leading-order~(NLO) QCD accuracy, that consistently include also the
fragmentation component, were obtained for the first time in
ref.~\cite{Binoth:1999qq}, and made publicly available in the {\tt DIPHOX}
software.
Next-to-next-to-leading-order~(NNLO) QCD corrections to $pp\rightarrow\gamma\gamma$
were obtained in
refs.~\cite{Catani:2011qz,Campbell:2016yrh,Catani:2018krb,Gehrmann:2020oec},
whereas NLO electroweak corrections have been studied in
ref.~\cite{Bierweiler:2013dja}. More recently, even the three-loop
$q\bar{q}\rightarrow \gamma\gamma$ amplitudes have been
computed~\cite{Caola:2020dfu}. In order to include all the effects up to
$\as^2$ accuracy, the gluon-induced production of a photon pair through a
closed quark loop has to be taken into account as well: such contribution,
that is particularly sizable due to the gluon luminosity, was first computed
in ref.~\cite{Dicus:1987fk}, and it is by now known at the next order, both
for massless~\cite{Bern:2002jx} and for massive quark
loops~\cite{Maltoni:2018zvp, Chen:2019fla}.  Substantial progress has also
been made for the computation of a photon pair in association with one or
more jets: predictions for diphoton in association with up to three jets,
including NLO QCD corrections, have been obtained in
refs.~\cite{DelDuca:2003uz,
  Gehrmann:2013aga, Gehrmann:2013bga, Badger:2013ava}, and NLO
electroweak~(EW) corrections for photon pair with up to two jets are also
known~\cite{Chiesa:2017gqx}. Very recently, the NNLO QCD results for $p p \rightarrow
\gamma\gamma + j$ were obtained as well~\cite{Chawdhry:2021hkp,
  Agarwal:2021vdh, Agarwal:2021grm}, and the two-loop amplitudes for $g g \rightarrow
\gamma \gamma + j$~\cite{Badger:2021imn}
were also computed.  Finally, as far
as all-order results are concerned, Sudakov logarithms of soft and collinear
origin, arising at all orders in the computation of the diphoton
transverse-momentum distribution, were resummed up to
next-to-next-to-leading-logarithmic~(NNLL) accuracy in
refs.~\cite{Balazs:2006cc, Nadolsky:2007ba, Balazs:2007hr, Cieri:2015rqa,
  Coradeschi:2017zzw}. More recently, results accurate up to N$^3$LL and
N$^3$LL$^\prime$ have also been obtained within the \Radish{}
formalism\footnote{This was shown in the plots in ref.~\cite{Alioli:2020qrd}
obtained with the \MatrixRadish{}
interface~\cite{Kallweit:2020gva}.}~\cite{Monni:2016ktx, Bizon:2017rah} and
the {\tt CuTe-MCFM} framework~\cite{Becher:2020ugp, Neumann:2021zkb},
respectively.

Nowadays, NLO Monte Carlo simulations for $pp\rightarrow\gamma\gamma$, matched to
parton showers~(PS), can be easily obtained with automated tools. A few
dedicated studies have been presented in the last decade: for instance, in
ref.~\cite{Hoeche:2009xc}, hard tree-level matrix elements with a variable
photon multiplicity were merged with a QCD+EW parton shower, allowing to take
simultaneously into account the direct and the fragmentation component. In
ref.~\cite{DErrico:2011cgc} these two components were also included in the
simulation, in the context of an exact matching of the NLO QCD corrections
with parton showers~(NLO+PS), using the {\sc\small Powheg}{}
method~\cite{Nason:2004rx}.\footnote{NLO+PS results for diphoton production,
including the $s$-channel exchange of a Kaluza-Klein resonance, were also
obtained in ref.~\cite{Frederix:2012dp}.}

Due to the increasing precision of experimental measurements and the fact
that NNLO QCD corrections to photon-pair production are known and sizable, it
is particularly important to match parton showers with the NNLO fixed order
calculation (NNLO+PS). A few methods have been developed to obtain NNLO+PS
predictions for color-singlet production at hadron colliders: ``NNLO
reweighted'' {\sc\small MiNLO}$^\prime$~\cite{Hamilton:2012rf, Hamilton:2013fea}
(sometimes also denoted  NNLOPS in the literature), {\tt
  Geneva}~\cite{Alioli:2012fc, Alioli:2013hqa},
{\tt UN$^2$LOPS}~\cite{Hoche:2014uhw}, and {\sc\small MiNNLO\textsubscript{PS}}{}~\cite{Monni:2019whf,
  Monni:2020nks}.\footnote{Ideas for going beyond NNLO+PS accuracy have been
discussed as well (see refs.~\cite{Frederix:2015fyz, Prestel:2021vww}), and very recently
NNLO+PS matching with sector showers, for final-state parton showers, has
been outlined in ref.~\cite{Campbell:2021svd}.}  Using these methods, many
processes, that at leading order~(LO) have only two external colored legs, have been described with NNLO+PS
accuracy~\cite{Hamilton:2013fea,
  Karlberg:2014qua, Astill:2016hpa, Astill:2018ivh, Re:2018vac,
  Bizon:2019tfo, Alioli:2015toa, Alioli:2019qzz, Alioli:2020fzf,
  Alioli:2020qrd, Alioli:2021qbf, Alioli:2021egp, Cridge:2021hfr,
  Hoche:2014uhw, Hoche:2014dla, Hoche:2018gti, Monni:2019whf, Monni:2020nks,
  Lombardi:2020wju, Lombardi:2021rvg, Hu:2021rkt,
  Buonocore:2021fnj}. Recently, results for top-pair production at NNLO+PS
were also published in ref.~\cite{Mazzitelli:2020jio, Mazzitelli:2021mmm}. As
far as diphoton production is concerned, the only NNLO+PS accurate results
have been obtained with the {\tt Geneva} method in
ref.~\cite{Alioli:2020qrd}.

In this paper we study diphoton production using the {\sc\small MiNNLO\textsubscript{PS}}{} method,
i.e.~we build a {\sc\small MiNLO}$^\prime$ simulation for $pp\rightarrow \gamma\gamma$ + jet
production, and we subsequently include NNLO QCD corrections according to the
procedure proposed in refs.~\cite{Monni:2019whf, Monni:2020nks}. The
resulting generator can be used to obtain NLO+PS accurate results for $\gamma\gamma j$
production, as well as to predict, with up to NNLO accuracy, observables that
are inclusive with respect to the presence of jets, such as the diphoton
invariant mass and rapidity, or the transverse momentum of the hardest and
next-to-hardest photon, retaining a consistent matching with parton showers.

Together with the main phenomenological results for diphoton
production, in the current paper we also describe a few novelties
introduced to deal with
the presence of
photons in the final state, but that could also be useful
for other (N)NLO+PS calculations.
First of all, already for the simulation of diphoton + jet production at
NLO+PS accuracy, one needs to deal with the fact that the Born-level $\gamma\gamma j$
matrix elements are divergent whenever a photon becomes soft or collinear to
a quark. In the following, we refer to these divergences as ``QED
divergences''.\footnote{Since no cuts are applied at the generation level, we
need to devise a treatment of QED-divergent regions during the generation of
the events, even though the final fiducial cuts will provide infrared-safe
results.}
We describe and implement a general way to deal with this issue in the {\sc\small Powheg}{}
formalism.\footnote{This method has already been used in
ref.~\cite{Lombardi:2020wju}, following the suggestion of some of us.}
Furthermore, the {\sc\small MiNNLO\textsubscript{PS}}{} method for the process at hand requires the
evaluation of the $q\bar{q}\rightarrow\gamma\gamma$ matrix-elements up to second
order in the strong coupling constant. Such matrix elements are divergent
when the photons are collinear to the beam axis: in order to avoid
introducing any cutoff or isolation criteria at any stage of the event
generation, we have also devised a new mapping from the $pp\rightarrow \gamma\gamma +
j$ to the $pp\rightarrow \gamma\gamma$ kinematics, that preserves the direction of
one photon with respect to the beam axis,
thereby allowing for a full control of the singular regions. The combined use
of the new mapping and of the method to deal with the QED divergences allowed
us to simulate diphoton production at NNLO+PS accuracy without introducing
any generation or technical cut, as done instead in
refs.~\cite{Lombardi:2020wju, Alioli:2020qrd}.
This gives us the
advantage that, on one side, we do not have to worry about checking the
independence of the fiducial physical differential cross sections from the
generation and technical cuts, and, on the other side, we can use the same
set of generated events for any choice of fiducial cuts.

In this paper we also propose a generalization of the {\sc\small MiNNLO\textsubscript{PS}}{} method that
allows for greater flexibility in the choice of the renormalization and
factorization scale used in the evaluation of the non-singular term for $F$ +
1 jet production, where $F$ is the color-singlet system.  In the original
formulation of the {\sc\small MiNNLO\textsubscript{PS}}{} method, such term is evaluated with scales of
the order of the transverse momentum of $F$.  The prescription we introduce
here generalizes such aspect of the {\sc\small MiNNLO\textsubscript{PS}}{} method, allowing one to treat
this term more similarly to a fixed-order computation. Such choice turned out
to be desirable in~$\gamma\gamma$ production, for comparisons with
fixed-order results, where this term gives an important contribution to the
total cross section, unlike in previous studies, such as those in
refs.~\cite{Monni:2019whf, Monni:2020nks}, where it was small.

The paper is organized as follows: in section~\ref{sec:outline} we describe
the ingredients used to build the event generator and the details of the
novel aspects we introduce here for the first time, whose validation is
discussed in section~\ref{sec:validation}. In section~\ref{sec:results} we
show some phenomenological results, and we compare our predictions with the
ATLAS diphoton data~\cite{ATLAS:2021mbt}. Finally we give our conclusions and
outlook in section~\ref{sec:conclusions}.

\section{Outline of the calculation}
\label{sec:outline}

In this section we review the theoretical formulation of our calculation.  We
first introduce the basic notation used throughout the paper and describe the
main theoretical issues to be dealt with in the implementation of a NNLO+PS
event generator for photon pair production. Then we discuss in detail the
handling of the QED singularities and introduce the mapping used to
project the $\Phi_{\gamma\gamma j}$ kinematics onto the $\PhiB$ one.  We
conclude the section with a description of the modifications we have
introduced in the {\sc\small MiNNLO\textsubscript{PS}}{} formalism in order to reproduce more accurately
the results of a fixed-order calculation for distributions totally inclusive
with respect to the QCD radiation.

\subsection{Description of the process}
\label{sec:implementation}
We consider the process of direct production of a photon pair in
proton-proton scattering
\begin{equation}
p p \,\rightarrow\, \gamma \gamma +X,
\end{equation}
with the requirement that the two photons are isolated, i.e.~each photon has
a minimum transverse momentum and is isolated with respect to the other
photon and to the final-state hadronic activity.  These requirements are
needed to make the process well-defined both from the theoretical and
the experimental point of view.
We call $p_{\gamma_1}$ and
$p_{\gamma_2}$ the momenta of the hardest and next-to-hardest photons and denote the
squared mass  and  transverse momentum
of the diphoton system as
\begin{equation}
  \label{eq:maa}
  Q^2 = m_{\gamma\gamma}^2 = \(p_{\gamma_1}+p_{\gamma_2}\)^2\!, \qquad \qquad
  p_{\sss T} = \(p_{\gamma_1}+p_{\gamma_2}\)_\tr.
\end{equation}
The starting point for the  {\sc\small MiNNLO\textsubscript{PS}}{} method we use in this work is the NLO
differential cross section for diphoton plus one jet production
\begin{equation}
pp \,\rightarrow\, \gamma\gamma  + j\,.
\end{equation}
The matrix elements for this process at NLO in QCD have
been obtained from the automated interface of the
{\sc\small Powheg Box Res}~\cite{Jezo:2016ujg} to \OpenLoopsTwo~\cite{Buccioni:2019sur,
  Cascioli:2011va, Buccioni:2017yxi, vanHameren:2010cp,
  vanHameren:2009dr}.

Since the goal of the  {\sc\small MiNNLO\textsubscript{PS}}{} formalism is to reach NNLO accuracy for
inclusive observables in the diphoton system, one also needs
the two-loop amplitudes for
$q\bar{q}\rightarrow\gamma\gamma$, that have  been taken from
refs.~\cite{Anastasiou:2002zn, Catani:2013tia} and implemented into the code.

In this paper, we work in the approximation of five light quarks and neglect
the contributions given by two-loop diagrams with the massive top quark. We
consider instead the contributions given by the massive top quark in the one-loop
diagrams for the process $pp \rightarrow \gamma\gamma + j$.

\subsection{Handling of the QED divergences}
\label{sec:QEDdiv}

In this section, we describe a general way to deal with processes that
present QED divergences at the Born level within the {\sc\small Powheg}{}
formalism.\footnote{The way the \PowhegBox{} deals with QCD divergences
at Born level has already been illustrated in several
papers,~e.g. ref.~\cite{Alioli:2010qp}.}
Before illustrating the method, we briefly review the relevant parts
of the \PowhegBox{} framework.

We start by introducing the general form of the {\sc\small Powheg}{}~\cite{Nason:2004rx}
differential cross section for a $2 \rightarrow n$ Born process, using the notation
of refs.~\cite{Frixione:2007vw, Alioli:2010xd}. Indicating with $\mathrm{d}\Phin$
the phase space for the $n$-body final state, we write the $(n+1)$-body phase
space $\mathrm{d}\Phinpo$ in terms of $\mathrm{d}\Phin$ and three additional
radiation variables, that we collectively label $\Phirad$
\begin{equation}
\mathrm{d}\Phinpo = \mathrm{d}\Phin \, \mathrm{d}\Phirad\,.
\end{equation}
We indicate with $B(\Phin)$, $V(\Phin)$ and $R(\Phinpo)$ the Born, virtual
and real amplitudes, convoluted with the corresponding parton distribution
functions (PDFs) and multiplied by the flux factor, and we split $R$ into two
positive contributions: $R_s(\Phinpo)$, that contains all the QCD
singularities, and a QCD-finite term, $R_f(\Phinpo)$, such
that\footnote{Please notice that $R_f(\Phinpo)$ could well be set to zero,
in which case $R_s(\Phinpo) = R(\Phinpo)$, the whole real contribution.}
\begin{equation}
\label{eq:Rs+Rf}
 R(\Phinpo) = R_s(\Phinpo) + R_f(\Phinpo).
\end{equation}
In general we achieve this separation with the help of a suitable function
$F(\Phinpo)$,
such that
\begin{equation}
R_s(\Phinpo) = F(\Phinpo) \, R(\Phinpo)\,, \qquad \qquad R_f(\Phinpo)
= \lq 1-F(\Phinpo)\rq R(\Phinpo)\,.
\end{equation}
The detailed form  for $F(\Phinpo)$ will be discussed in section~\ref{sec:F}.
When $R(\Phinpo)$ is split as in eq.~(\ref{eq:Rs+Rf}),
the {\sc\small Powheg}{} differential cross section can be written as
\begin{eqnarray}
\label{eq:PWG_sig}
\mathrm{d}\sigma &=& \bar{B}\!\(\Phin \)\!  \lg \Delta\!\(\Phin,\ensuremath{k_{\sss\rm T}^{\rm \sss min}}\)
+ \theta\!\({\ensuremath{k_{\rm\sss T}}\xspace}-\ensuremath{k_{\sss\rm T}^{\rm \sss min}}\) \Delta\!\(\Phin,{\ensuremath{k_{\rm\sss T}}\xspace}\) \, \frac{{
    R_s\!\(\Phinpo \)}}{B\!\(\Phin \)} \, \,\mathrm{d} \Phirad \rg
\mathrm{d}\Phin
\nonumber \\
&& {} + R_f\!\(\Phinpo \) \mathrm{d}\Phinpo\,,
\end{eqnarray}
where ${\ensuremath{k_{\rm\sss T}}\xspace}$ is the transverse momentum of the {\sc\small Powheg}{} jet,
\begin{equation}
\label{eq:Bbar}
\bar{B}(\Phin) =  B(\Phin) + V(\Phin) + \int \mathrm{d} \Phirad \, { R_s(\Phin,
  \Phirad)}\,,
\end{equation}
and the {\sc\small Powheg}{} Sudakov form factor
\begin{equation}
\label{eq:Sudakov_PWG}
 \Delta\!\(\Phin, {\ensuremath{k_{\rm\sss T}}\xspace}\) = \exp \lg - \int \mathrm{d} \Phirad'\,
   { \frac{{ R_s(\Phin, \Phirad')}}{B(\Phin)}} \,  \theta\!\lq
   {\ensuremath{k_{\rm\sss T}}\xspace}'\!\(\Phin,\Phirad'\) - {\ensuremath{k_{\rm\sss T}}\xspace} \rq   \rg
\end{equation}
contains only the QCD singular part of the real contributions
$R_s(\Phinpo)$. In the expressions above $\ensuremath{k_{\sss\rm T}^{\rm \sss min}}$ acts as an infrared cutoff
for unresolved radiation.

If the Born amplitude is divergent, the \PowhegBox{} applies a suppression
factor $\SB(\Phin)$ to $\bar{B}(\Phin)$ such that the product $\SB(\Phin)
\bar{B}(\Phin)$ is integrable over the entire Born phase space, and the
$\Phin$ kinematics can be generated from the distribution $\SB(\Phin)\bar{B}(\Phin)$
without the need of any hard cut. At the end, each event is given a weight
$1/\SB(\Phin)$ in order to compensate for the suppression factor.

In general, this way of regularizing divergences by means of a suppression
factor that depends on $\Phin$ can be used every time QCD and/or QED
singularities are present at the Born level.
While applying a suppression factor on $\bar{B}(\Phin)$ is enough when only
QCD singularities are present, for diphoton production we also have to deal with the QED
singularities appearing inside $R_f(\Phinpo)$.
We cannot simply suppress them through a second suppression factor
(that would be a function of $\Phinpo$) since this term is integrated
in the radiation variables inside $\bar{B}(\Phin)$, preventing one to compensate for such suppression factor, after the events have been generated.

We then proceed as follows: we take advantage of the possibility to
separate the real contributions into two terms (see
eq.~(\ref{eq:Rs+Rf})) in such a way that $R_s(\Phinpo)$ contains all the QCD
singularities but no QED ones.
Since the generation of $\Phinpo$ according to $R_f(\Phinpo)$ in
eq.~(\ref{eq:PWG_sig}) is performed with a hit-and-miss technique, we apply a
QED suppression factor $\SR(\Phinpo)$ to $R_f(\Phinpo)$, and generate
$\Phinpo$ according to the product $\SR(\Phinpo) R_f(\Phinpo)$, that is now
integrable. Finally, we multiply the weights of the generated events by the
factor $1/\SR(\Phinpo)$.\footnote{To the best of our knowledge, a similar
procedure was used for the first time in ref.~\cite{DErrico:2011cgc}.}

\subsubsection[The damping function $F$]{The damping function $\boldsymbol{F}$}
\label{sec:F}

According to the FKS method~\cite{Frixione:1995ms, Frixione:1997np}, the real
contribution $R$ is partitioned into a sum of terms $R^{\alr}$, each of them
having at most one  collinear and one soft singularity associated with one
parton

\begin{equation}
 R = \sum_{\alr} R^\alr\,.
 \end{equation}
 %
In each $\alr$ region, the radiated parton~(r) can then become soft or be collinear to an emitting
one~(e), and we can define the damping function
\begin{equation}
F^\alr = \frac{
  \(  \frac{\displaystyle 1}{\displaystyle d_{\rm\sss  e,r} }  \)^p
  }
  {\displaystyle
 \(  \frac{\displaystyle 1}{\displaystyle d_{\rm\sss  e,r}}  \)^p
   +  \sum_{i = 1}^{n_{\rm c}} \sum_{j=1}^{n_{\gamma}}
  \( \frac{1}{d_{c_i,\gamma_j}} \)^p }\,,
\end{equation}
where
\begin{equation}
  \label{eq:dij}
  d_{i,j} = \lg
  \begin{array}{l l}
    \displaystyle p_{\tr j}^2& \mbox{if $i$ is an initial-state
      particle,}
    \\[2mm]
    \displaystyle 2\min\!\(E_i^2,E_j^2\) \(1-\cos\theta_{ij}\) \qquad &
    \mbox{if $i$ and $j$ are final-state particles,}
  \end{array}
  \right.
\end{equation}
and the sum in the denominator runs over the $n_{\sss\rm c}$ massless charged
particles and the $n_{\gamma}$ photons. In eq.~(\ref{eq:dij}), $p_{\tr j}$
and $E_j$ are the transverse momentum and energy of the particle $j$, and
$\theta_{ij}$ the angle between the particles $i$ and $j$, computed in the
partonic center-of-mass frame.\footnote{In our simulation, we have set
$p=2$.}

Every contribution $R^\alr$ to the real amplitude is then split into
two terms
\begin{equation}
R^{\alr} = R^{\alr}_{s} + R^{\alr}_{f}\,,
\end{equation}
where
\begin{equation}
R^{\alr}_{s} \equiv F^{\alr} R^{\alr}, \qquad \qquad\qquad R^{\alr}_{f}
\equiv \( 1- F^{\alr} \) R^\alr\,.
\end{equation}
In the limit where the radiated parton is soft or collinear to the emitter
$d_{\rm\sss e,r}$ is small, and $F^{\alr}$ tends to 1, so that all the QCD
singularities are in $R^{\alr}_{\sss\rm s}$.  When instead the photon
$\gamma_j$ becomes soft or collinear to a massless charged particle $c_i$,
the term $d_{c_i,\gamma_j}$ is small, and $F^{\alr}$ tends to 0, so that
$R^{\alr}_{\sss\rm s}$ is free from QED singularities.

\subsubsection[The suppression factors $\SB$ and
  $\SR$]{The suppression factors $\boldsymbol{\SB}$ and
  $\boldsymbol{\SR}$}
\label{sec:S_B-S_R}
We have chosen as  suppression factor $\SB(\Phin)$ introduced in
section~\ref{sec:QEDdiv} the following expression
\begin{eqnarray}
  \label{eq:supp_fact_Born}
\SB &=& \frac{\( p^2_{\tr j}\)^a}{ \( p^2_{\tr j}\)^a + \(\bar{p}^2_{\tr
    j}\)^a}
\,\frac{\( p^2_{\tr \gamma_1}\)^a}{ \( p^2_{\tr \gamma_1}\)^a + \(\bar{p}^2_{\tr \gamma}\)^a}
\,\frac{\( p^2_{\tr \gamma_2}\)^a}{ \( p^2_{\tr \gamma_2}\)^a +
  \(\bar{p}^2_{\tr \gamma}\)^a}
\nonumber\\[2mm]
&& {} \times
\frac{\( R^2_{j \gamma_1}\)^a}{ \( R^2_{j \gamma_1}\)^a + \(\bar{R}^2_{j \gamma}\)^a}
\,\frac{\( R^2_{j \gamma_2}\)^a}{ \( R^2_{j \gamma_2}\)^a + \(\bar{R}^2_{j \gamma}\)^a},
\end{eqnarray}
where $p_{\tr i}$ is the transverse momentum of the particle $i$ with
respect to the beam axis, $R_{ij}$ the angular distance between
the particles $i$ and $j$ in the azimuth-pseudorapidity plane
\begin{equation}
    R_{ij} =\sqrt{\(\eta_i-\eta_j\)^2+\(\phi_i-\phi_j\)^2},
\end{equation}
and the barred quantities are arbitrary constant parameters needed to define
the transition point (from zero to one) of each factor in the product.  The
first three term of eq.~(\ref{eq:supp_fact_Born}) suppress the limit where
the final-state parton or the two photons are soft or collinear to the beam
axis, while the last two terms suppress the regions where the photons are
collinear to the final-state parton. The power $a$ does not need to be the
same for all the terms, but we have chosen the common value $a = 1$ for
simplicity. The expression for $\SR(\Phinpo)$ can be obtained by generalizing
eq.~(\ref{eq:supp_fact_Born}), for the case with an extra jet.

When the {\sc\small MiNNLO\textsubscript{PS}}{} formalism is used, the QCD initial-state singularity is
already regularized by the Sudakov form factor from the resummation
formalism, and the first term in eq.~(\ref{eq:supp_fact_Born}) is no longer
needed. We give more details in section~\ref{sec:S_B-S_R_MiNNLO}.

\subsection[The $\PhiBj \rightarrow \PhiB$ mapping]{The $\boldsymbol{\PhiBj} \rightarrow \boldsymbol{\PhiB}$ mapping}
\label{sec:mapping}

When applying the {\sc\small MiNNLO\textsubscript{PS}}{} formalism, we need a mapping from the
$\gamma\gamma j$ phase space to the $\gamma\gamma$ one. In fact, as it will
be recalled in the next section, the Sudakov form factor (eq.~(\ref{eq:Sud}))
and the $D$ term (eq.~(\ref{eq:Dterms})), at the core of the {\sc\small MiNNLO\textsubscript{PS}}{}
method, are functions of $\PhiB$, as they contain the
$q\bar{q}\rightarrow\gamma\gamma$ amplitudes. One needs then to evaluate these quantities
while integrating over the $\Phi_{\gamma\gamma j}$ phase space upon which the
{\sc\small Powheg}{} $\bar{B}$ function depends, thereby requiring a smooth
$\Phi_{\gamma\gamma j}\rightarrow \PhiB$ mapping.

The $q\bar{q}\rightarrow\gamma\gamma$ amplitudes are singular when the photons are
collinear to the beam axis. From the physics point of view, one does not
expect to evaluate such amplitudes arbitrarily close to their singularities,
as such phase space points are discarded by the request of the presence of
two isolated photons in the final state.  Since in the {\sc\small MiNNLO\textsubscript{PS}}{} formalism
the $\PhiB$ kinematics is obtained by a mapping from the $\Phi_{\gamma\gamma
  j}$ one, we need to make sure that we never get too close to the singular
regions, and without having to introduce an explicit cut on the transverse momentum of
the single photons in the $\PhiB$ kinematics.

We have built such a mapping without having to introduce any cutoff or
isolation criteria, at any stage of the event generation, as done in other
Monte Carlo simulations dealing with photons~\cite{Alioli:2020qrd,
  Lombardi:2020wju}.
The mapping that we propose is such that the direction of one of the photons
with respect to the beam axis in the laboratory frame, for a given phase space
point $\Phi_{\gamma\gamma j}$, is preserved in the projected point $\PhiB$. As
a consequence, a point in $\PhiB$ with small $p_{\sss T}$ always comes from a
projection of a point in $\Phi_{\gamma\gamma j}$ where at least one photon is
close to the beam axis, and this configuration is suppressed by the factor
$\SB$ of eq.~(\ref{eq:supp_fact_Born}).
A detailed derivation of such a mapping is given in
appendix~\ref{app:projection}.

\subsection{{\sc\small MiNNLO\textsubscript{PS}}{} differential cross section}
\label{sec:MiNNLO}
In this section, after briefly recalling the general method, we
discuss the modifications that we have introduced in the definition of
the {\sc\small MiNNLO\textsubscript{PS}}{} differential cross section given in
refs.~\cite{Monni:2019whf, Monni:2020nks}.
Following the conventions introduced in those papers,
we write the $p_{\sss T}$ spectrum of the cross section
for diphoton production as
\begin{equation}
\label{eq:sig_s_f}
  \frac{\mathrm{d}\sigma}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} =
  \frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} +
  \frac{\mathrm{d}\sigma_{\sss \! f}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}},
\end{equation}
where $\sigma_{r}$ (also called singular contribution in the following)
is obtained from the resummation of logarithmically enhanced
contributions at small-$p_{\sss T}$, while $\sigma_{\sss \! f}$ (also called non-singular term) is the
difference between the fixed-order differential cross section
and the truncated
perturbative $\as$ expansion of $\sigma_{r}$, till the second order.
The singular contribution can be written as
\begin{equation}
  \label{eq:sig_s}
  \frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} = \frac{\mathrm{d}}{\mathrm{d}
    p_{\sss T}} \lg \exp\!\lq -\tilde{S}\!\(\PhiB,p_{\sss T}\) \rq \! {\cal L}\!\(\PhiB,
  p_{\sss T}\)\rg,
\end{equation}
where the Sudakov form factor is given by
\begin{equation}
  \label{eq:Sud}
  \tilde{S}\!\(\PhiB, p_{\sss T}\) = 2\int_{p_{\sss T}}^{Q}\frac{\mathrm{d} q}{q}
  \lg  A\Big(\! \as\!\(q\)\!\Big) \ln\frac{Q^2}{q^2} +
  \tilde{B}\Big(\! \as\!\(q\)\!\Big) \rg
\end{equation}
and ${\cal L}$ is a function of the PDFs, of the $q \bar{q} \rightarrow \gamma
\gamma$ Born, one- and two-loop matrix elements, and of the collinear
coefficient functions (see ref.~\cite{Catani:2012qa}). The functions
$A$ and $\tilde{B}$ in eq.~(\ref{eq:Sud}) can be written as
\begin{eqnarray}
  A(\as) &=& \left(\frac{\as}{2\pi}\right) A^{(1)} + \left(\frac{\as}{2\pi}\right)^2
  A^{(2)}+ \left(\frac{\as}{2\pi}\right)^3 A^{(3)}\,,
  %
  \\
  %
  \tilde{B}(\as) &=& \left(\frac{\as}{2\pi}\right) B^{(1)} + \left(\frac{\as}{2\pi}\right)^2
  \tilde{B}^{(2)}\,,
\end{eqnarray}
with
\begin{equation}
  \tilde{B}^{(2)} = B^{(2)} + \beta_0 H^{(1)} + 2\zeta_3\(A^{(1)}\)^2,
\end{equation}
where $H^{(1)}$ is the ratio between the one-loop and the Born $q \bar{q} \rightarrow
\gamma \gamma$ amplitudes in the \ensuremath{\overline{\rm MS}}{} subtraction scheme, and introduces a
dependence on the phase-space kinematics in $\tilde{B}^{(2)}$. The
coefficients $A^{(1)}$, $A^{(2)}$, $A^{(3)}$, $B^{(1)}$, and $B^{(2)}$ for
quark-initiated processes are collected, for example, in
ref.~\cite{Monni:2019whf}. More details can be found in
appendix~\ref{app:scale_dep}.

The non-singular contribution $\mathrm{d}\sigma_{\sss \! f}$ is instead given by the
difference
\begin{equation}
  \label{eq:sig_f}
  \frac{\mathrm{d}\sigma_{\sss \! f}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} =
\left.
\frac{\mathrm{d}\sigma^{\rm \sss NLO}_{\gamma\gamma j}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}
\right|_Q
- \abarmu{Q} \!
\left[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}\right]^{(1)}_Q
- \left(\abarmu{Q}\right)^{\!2}\!
\left[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}\right]^{(2)}_Q,
\end{equation}
where $\mathrm{d}\sigma^{\rm \sss NLO}_{\gamma\gamma j}$ is the NLO
differential cross section for $\gamma\gamma j$ production, and $\lq
\mathrm{d}\sigma_{r} \rq^{(i)}$ is the $i$-th order of the expansion of
$\mathrm{d}\sigma_{r}$ in the strong coupling constant. In this paper, we use
the notation $[X]^{(i)}$ for the coefficient of the $i$-th term in the
perturbative expansion of the quantity $X$. The above difference is
integrable, since the expansion of $\sigma_s$ cancels the
non-integrable terms of $\mathrm{d}\sigma^{\rm \sss NLO}_{\gamma\gamma j}$
in the $p_{\sss T} \rightarrow 0$ limit.
At variance with refs.~\cite{Monni:2019whf, Monni:2020nks,
  Lombardi:2020wju, Lombardi:2021rvg, Mazzitelli:2020jio}, we have
chosen to set the renormalization and factorization scales $\mur$ and
$\muf$ in $\mathrm{d}\sigma_{\sss \! f}$ to $Q$, instead of $p_{\sss T}$. This is why we
have added the subscript $Q$ to the quantities appearing in
eq.~(\ref{eq:sig_f}). While formally any scale choice would be
legitimate for evaluating this term, since the difference would be of
order ${\cal O}(\as^3)$, beyond the accuracy of our calculations,
setting the scale to $Q$ allows to follow more closely what is
typically adopted in fixed-order calculations, thereby allowing for a
more accurate comparison with the latter.
Moreover, the choice of the central scale also plays a role in the
estimation of the theoretical uncertainties. In this case, since the
size of the non-singular contribution is numerically relevant, setting
the scale to $p_{\sss T}$ would result in an unnatural enhancement of the
scale-variation bands.
We will give more quantitative details in
section~\ref{sec:DY_and_Higgs}, where we compare the new $\mu = Q$ and the
$\mu = p_{\sss T}$ scale choices for $\mathrm{d}\sigma_{\sss \! f}$ for the previously
discussed cases of Drell-Yan and Higgs boson production.

Following refs.~\cite{Monni:2019whf, Monni:2020nks}, the singular
contribution of eq.~(\ref{eq:sig_s}) can be rewritten as
\begin{equation}
\label{eq:expand_diff}
  \frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} = \exp\!\lq
  -\tilde{S}\!\(\PhiB, p_{\sss T}\) \rq \! D\!\(\PhiB, p_{\sss T}\),
\end{equation}
where
\begin{equation}
\label{eq:Dterms}
D\!\(\PhiB, p_{\sss T}\)  \equiv -\frac{\mathrm{d} \tilde{S}\!\(\PhiB,p_{\sss T}\)}{\mathrm{d} p_{\sss T}} \, {\cal L}\!\(\PhiB, p_{\sss T}\)
+ \frac{\mathrm{d} {\cal L}\!\(\PhiB, p_{\sss T}\)}{\mathrm{d} p_{\sss T}}\,.
\end{equation}
By expanding eq.~(\ref{eq:expand_diff}) in $\as(Q)$, we can write
\begin{eqnarray}
\label{eq:dsigs1}
\left[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}\right]^{(1)}_Q &=&
\Big[ D\!\(\PhiB, p_{\sss T}\) \Big]^{(1)}_Q
\\
\label{eq:dsigs2}
\left[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}\right]^{(2)}_Q &=&
\Big[ D\!\(\PhiB, p_{\sss T}\) \Big]^{(2)}_Q - \lq \tilde{S}\!\(\PhiB,p_{\sss T}\) \rq^{(1)}
\Big[ D\!\(\PhiB, p_{\sss T}\) \Big]^{(1)}_Q
\end{eqnarray}
where all the terms are evaluated at $\mur=\muf=Q$.

The $D\!\(\PhiB, p_{\sss T}\)$ term in eq.~(\ref{eq:expand_diff}) has a
formal expansion
\begin{eqnarray}
\label{eq:D_expans_pt}
D\!\(\PhiB, p_{\sss T}\) &=&
\frac{\as(p_{\sss T})}{2\pi} \lq D\!\(\PhiB, p_{\sss T}\) \rq^{(1)} +
\(\frac{\as(p_{\sss T})}{2\pi}\)^2 \lq D\!\(\PhiB, p_{\sss T}\) \rq^{(2)}
\nonumber\\
&& \mbox{} +
\(\frac{\as(p_{\sss T})}{2\pi}\)^3 \lq D\!\(\PhiB, p_{\sss T}\) \rq^{(3)} + \ldots
\end{eqnarray}
and, at difference with eqs.~(\ref{eq:dsigs1}) and~(\ref{eq:dsigs2}), all
the terms are evaluated at $\mur=\muf=p_{\sss T}$.\footnote{We stress the fact that
we do not compute separately the $\lq D\!\(\PhiB, p_{\sss T}\) \rq^{(i)}$ terms, but
we compute numerically the whole function $D$, as discussed, for the
first time, in ref.~\cite{Monni:2020nks}.}
The explicit expression of these terms are collected in
appendix~\ref{app:scale_dep}, together with the expressions of the other
ingredients needed in the {\sc\small MiNNLO\textsubscript{PS}}{} formulae.

We would like to stress that the cancellation of the divergences associated
with the small $p_{\sss T}$ limit in eq.~(\ref{eq:sig_f}) is numerically
challenging.
For this reason, and following the {\sc\small MiNLO}{} original approach,  we have chosen to
multiply $\mathrm{d}\sigma_{r}$ by a Sudakov form factor, after adding an
additional term to preserve the nominal $\as^2$ accuracy
\begin{equation}
\label{eq:sigf_times_sud}
\frac{\mathrm{d}\sigma_{\sss \! f}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} \, \rightarrow \, \exp\!\lq
-{\bar{S}}\!\(p_{\sss T}\) \rq \lg
\frac{\mathrm{d}\sigma_{\sss \! f}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}} +
\(\frac{\as(Q)}{2\pi}\)^2 \lq {\bar{S}}\!\(p_{\sss T}\) \rq^{(1)} \lq
\frac{\mathrm{d}\sigma_{\sss \! f}}{\mathrm{d}\PhiB\mathrm{d} p_{\sss T}}\rq^{(1)} \rg.
\end{equation}
At difference with~\cite{Monni:2019whf, Monni:2020nks,
  Lombardi:2020wju, Lombardi:2021rvg, Mazzitelli:2020jio}, where
$\bar{S}=\tilde{S}$, we apply a Sudakov
form factor with only the two leading terms, i.e.
\begin{equation}
  \bar{S}\!\(p_{\sss T}\) = 2 \int_{p_{\sss T}}^{Q} \frac{dq}{q}
  \frac{\as\!\(q\)}{2\pi} \lq  A_1\log\!\(\frac{Q^2}{q^2}\) +
    B_1 \rq.
\end{equation}
We stress that the two approaches are equivalent up to $\mathcal{O}{\as^2}$.

Summarizing, the {\sc\small MiNNLO\textsubscript{PS}}{} differential
cross section for $pp\rightarrow \gamma\gamma + X$ production can be written
as
\begin{eqnarray}
\label{eq:dsig_final}
\frac{\mathrm{d}\sigma_{\sss \gamma\gamma j}}{\mathrm{d}\PhiBj}
&=&
\exp\!\lq -\tilde{S}\!\(\PhiB,p_{\sss T}\) \rq \! D\!\(\PhiB, p_{\sss T}\)  F^{\tmop{corr}}\!\(\PhiBj\)
\nonumber\\
&+&
\exp\!\lq -{\bar{S}}\!\(p_{\sss T}\) \rq
\!\Bigg\{\!
\left.
\frac{\mathrm{d}\sigma^{\rm \sss NLO}_{\gamma\gamma j}}{\mathrm{d}\PhiBj}
\right|_Q
+
 \(\frac{\as(Q)}{2\pi}\)
 \lq {\bar{S}}\!\(p_{\sss T}\) \rq^{(1)}
 \left.
 \frac{\mathrm{d}\sigma^{\rm \sss  LO}_{\gamma\gamma j}}{\mathrm{d}\PhiBj}
\right|_Q
\nonumber\\
&&
\mbox{}\hspace{15mm}
- \lq \abarmu{Q} \!
\Big[ D\!\(\PhiB, p_{\sss T}\)\! \Big]^{(1)}_Q
+ \left(\abarmu{Q}\right)^{\!2}\!
\Big[ D\!\(\PhiB, p_{\sss T}\)\! \Big]^{(2)}_Q \rq \!F^{\tmop{corr}}\!\(\PhiBj\)
\!\!\Bigg\},
\nonumber\\
\end{eqnarray}
where we have introduced the symbol ${\mathrm{d}\sigma^{\rm \sss LO}_{\gamma\gamma
    j}}/{\mathrm{d}\PhiBj}$ to denote the LO differential cross section for $p p \rightarrow
\gamma \gamma j$ production,
and the function $F^{\tmop{corr}}\!\(\PhiBj\)$, needed to spread the contributions
proportional to the $D(\PhiB, p_{\sss T})$ terms over the entire $\PhiBj$ phase
space. It has the property that, given an arbitrary function
$G\!\(\PhiB,p_{\sss T}\)$,
\begin{equation}
\int \mathrm{d}\PhiBj \, G\!\(\PhiB,p_{\sss T}\) F^{\tmop{corr}}\!\(\PhiBj\) = \int
\mathrm{d}\PhiB \, \mathrm{d}p_{\sss T} \, G\!\(\PhiB,p_{\sss T}\).
\end{equation}
For the explicit expression we have used for $F^{\tmop{corr}}\!\(\PhiBj\)$ we refer to
ref.~\cite{Monni:2019whf}.

Equation~(\ref{eq:dsig_final}) is the main result of this section and it
summarizes the novel aspects we introduced to the {\sc\small MiNNLO\textsubscript{PS}}{} method,
as can be evinced by comparing it against,~e.g., eq.~(3.7) of
ref.~\cite{Monni:2019whf}. In the rest of the manuscript, we will
denote the {\sc\small MiNNLO\textsubscript{PS}}{} results obtained according to
eq.~(\ref{eq:dsig_final}) with the acronym {\tt FOatQ}. We also
recall that the mapping introduced in section~\ref{sec:mapping}
guarantees that all the $\PhiB$-dependent terms in
eq.~(\ref{eq:dsig_final}) are
evaluated in kinematic points away from the singular regions.

By rewriting the NLO differential cross section for $\gamma\gamma j$
production in the following compact form
\begin{equation}
  \mathrm{d}\sigma^{\rm \sss NLO}_{\gamma\gamma j} = \(B + V \) {\mathrm{d}\PhiBj} +
  \(R_{\sss s} + R_{\sss f}\){\mathrm{d}\PhiBjj}\,,
\end{equation}
and introducing the  {\sc\small MiNNLO\textsubscript{PS}}{}
improved $\bar{B}$ function
\begin{eqnarray}
  \bar{B}\!\(\PhiBj\)_{\rm\sss MiNNLO_{PS}}\! &=&\!
 \exp\!\lq -{\tilde{S}}\!\left(\PhiB, p_{\sss T}\right)\rq
 D\!\(\PhiB, p_{\sss T}\) F^{\tmop{corr}}\!\(\PhiBj\)
\nonumber \\
&& {}\! + \exp\!\lq -{\bar{S}}\!\left(p_{\sss T}\right)\rq \Bigg\{\!\!
\lq 1 + \frac{\as(Q)}{2\pi}  \lq {\bar{S}}\!\left(p_{\sss T}\right) \rq^{(1)}
\rq  B \big|_Q     +  V\big|_Q  +
\int \mathrm{d}\Phi_\rad \,  R_s\big|_Q
\nonumber \\
&& {}\! - \left[\! \left(\frac{\as(Q)}{2\pi}\right) \!  \Big[ D\!\(\PhiB,
    p_{\sss T}\)\! \Big]^{(1)}_Q + \left(\abarmu{Q}\right)^{\!2}\!  \Big[
    D\!\(\PhiB, p_{\sss T}\)\! \Big]^{(2)}_Q \right] \!F^{\tmop{corr}}\!\(\PhiBj\)\!\! \Bigg\},
\nonumber\\
\end{eqnarray}
our final expression for  the differential cross section $\mathrm{d}\sigma_{\sss
  \rm \gamma\gamma}$ reads
\begin{eqnarray}
  \label{eq:MiNNLOPS+Powheg_formula}
  \mathrm{d}\sigma_{\sss \rm \gamma\gamma} &=&  \bar{B}\!\(\PhiBj\)_{\rm\sss MiNNLO_{PS}}
  \Bigg\{ \Delta\!\left(\PhiBj, \ensuremath{k_{\sss\rm T}^{\rm \sss min}}\right)
\mathrm{d}\PhiBj  +  \theta\!\left({\ensuremath{k_{\rm\sss T}}\xspace}- \ensuremath{k_{\sss\rm T}^{\rm \sss min}}\right)
\Delta\!\left(\PhiBj,{\ensuremath{k_{\rm\sss T}}\xspace}\right)
\left. \frac{R_{\sss s}}{B}\right|_{\ensuremath{k_{\rm\sss T}}\xspace} \mathrm{d}\PhiBjj \Bigg\}
\nonumber \\
&& {} + R_{\sss f} \big|_Q \, \mathrm{d}\PhiBjj\,.
\end{eqnarray}

\subsubsection{The suppression factors with {\sc\small MiNNLO\textsubscript{PS}}{}}
\label{sec:S_B-S_R_MiNNLO}
As already discussed in section~\ref{sec:S_B-S_R}, we do not need to suppress
the small $p_{\sss T}$ region of the first jet while the {\sc\small Powheg Box Res}{} integrates the
$\bar{B}\!\(\PhiBj\)_{\rm\sss MiNNLO_{PS}}$ function over the whole phase
space, since the presence of the {\sc\small MiNNLO\textsubscript{PS}}{} Sudakov form factor suppresses
such region.
In this case, the Born suppression
factor $\SB$ in eq.~(\ref{eq:supp_fact_Born}) can be replaced by
\begin{equation}
  \label{eq:supp_fact_Born_MiNNLO}
S_{\sss \rm B}^{\rm\sss MiNNLO} = \frac{\( p^2_{\tr \gamma_1}\)^a}{ \( p^2_{\tr \gamma_1}\)^a + \(\bar{p}^2_{\tr \gamma}\)^a}
\,\frac{\( p^2_{\tr \gamma_2}\)^a}{ \( p^2_{\tr \gamma_2}\)^a +
  \(\bar{p}^2_{\tr \gamma}\)^a}
\frac{\( R^2_{j \gamma_1}\)^a}{ \( R^2_{j \gamma_1}\)^a + \(\bar{R}^2_{j \gamma}\)^a}
\,\frac{\( R^2_{j \gamma_2}\)^a}{ \( R^2_{j \gamma_2}\)^a + \(\bar{R}^2_{j \gamma}\)^a}\,.
\end{equation}

\subsubsection{Scale settings in the small-$p_{\sss T}$ limit and modified logarithms}
\label{sec:scales}

We freeze the renormalization and factorization scales at values below $Q_0 =
2$~GeV to avoid the Landau pole and further non-perturbative effects,
connected with the PDF evolution to lower scales.
We stress that this scale does not act as a cutoff in the integration over
the physical space
but
only enters in the evaluation of the singular
contribution in eq.~(\ref{eq:expand_diff}), since all the other terms
in our formulation of the {\sc\small MiNNLO\textsubscript{PS}}{} differential cross section are evaluated
at the hard scale $Q$.
We also highlight that, due to the presence of an overall Sudakov form
factor, the dependence of the differential cross
section on $Q_0$ is strongly suppressed.\footnote{
We have found no visible dependence on $Q_0$ of our results by lowering its value
down to values of 1~GeV.}

In addition, following what was done in ref.~\cite{Monni:2019whf, Monni:2020nks}, we
smoothly turn off the contribution of the logarithms in the $D$ functions at
large transverse momentum with the replacement
\begin{equation}
  \log\frac{Q}{p_{\sss T}} \, \rightarrow \, \frac{1}{p} \log\!\( 1 + \(\frac{Q}{p_{\sss T}}\)^p\),
  \end{equation}
so that the $p_{\sss T} \rightarrow 0$ limit remains unaffected, while at $p_{\sss T} > Q$, the
modified logarithm tends to zero. In our simulation we have set
$p=6$.\footnote{We have explicitly checked that the phenomenological results
we present in section~\ref{sec:results} are independent of the exact value of $p$. In fact, we
have changed this value in the range from 1 to 9 without obtaining any appreciable
difference in the final results.}

\section{Validation}
\label{sec:validation}
In this section we discuss the validation of our {\sc\small MiNNLO\textsubscript{PS}}{} predictions.  We
first briefly present the setting of the physical and technical parameters of
the calculation and the isolation criterion used to define the diphoton
process.  We then study the effects that the modifications to the {\sc\small MiNNLO\textsubscript{PS}}{}
differential cross section described in section~\ref{sec:MiNNLO} have on two
previous implementations of the method (i.e.~the Drell-Yan and Higgs
production processes). Finally, we present a validation of the {\sc\small MiNNLO\textsubscript{PS}}{}
results for diphoton production against the fixed-order distributions
produced with the public version of the \Matrix{} code.

\subsection{Physical and technical parameters and photon isolation criterion}
\label{sec:physical_parameters}

The phenomenological results presented in this paper have been
obtained for a proton-proton collider with a hadronic center-of-mass
energy of 13~TeV. We have used the \LHAPDF~\cite{Buckley:2014ana}
parton distribution function set \verb|NNPDF31_nnlo_as_0118| and the
evolution of $\as$ provided by the same package. The electromagnetic
coupling for the final-state photons has been set to $\alpha =
{1}/{137}$, and the mass of the top quark to $m_{\sss\rm top} =
173.2~{\rm GeV}$.

The fixed-order results have been obtained with the public
  version of the
\Matrix{} code~\cite{Grazzini:2017mhc, Catani:2011qz,
  Cascioli:2011va,Denner:2016kdg, Anastasiou:2002zn, Catani:2012qa,
  Catani:2007vq}, setting the central renormalization and
factorization scales equal to the invariant mass of the diphoton
system $\mur = \muf = Q$. The uncertainty band has been estimated
via a seven-point scale variation obtained by
multiplying and dividing the central renormalization and factorization scales by a
factor of 2.

We apply the photon
isolation prescription of ref.~\cite{Frixione:1998jh} to the two
final-state photons. For each photon, we compute the angular distance
$R_{i\gamma}$ with respect to the $i$-th final-state parton. We
discard the event unless, for every photon and every $R<R_c$,
\begin{equation}
  \label{eq:fx_isolation}
  \sum_{i=1}^{n_{\rm \sss i}} \pti  \, \theta \!\(R-R_{i\gamma}\) < {E_\tr^\MAX} \,  \chi(R),
\end{equation}
where $n_{\rm\sss i}$ is the number of final-state partons, $\pti$ is
the transverse momentum of $i$ with respect to the beam, and
\begin{equation}
  \chi(R) = \( \frac{1 - \cos R}{1 - \cos {R_{\sss\rm c}}}\)^n\!.
\end{equation}
In our analysis, we have set
\begin{equation}
  {E_\tr^\MAX} = 4~{\rm GeV}, \qquad\qquad {R_{\sss\rm c}} = 0.4\,, \qquad\qquad  n=1\,.
\end{equation}
In addition, the two photons have to fulfill
\begin{equation}
  \label{eq:kinematic_cuts}
  {p_{\tr\gamma_1}} > {p_{\tr\gamma_1}^\MIN}\,,
  %
  \qquad\qquad
  %
  {p_{\tr\gamma_2}} > {p_{\tr\gamma_2}^\MIN}\,,
  %
  \qquad\qquad
  %
  {m_{\gamma\gamma}} > {m_{\gamma\gamma}^\MIN}\,,
\end{equation}
where ${p_{\tr\gamma_1}}$ and ${p_{\tr\gamma_2}}$ are the transverse momenta of the
hardest and next-to-hardest photons,  ${m_{\gamma\gamma}}$ is the diphoton
mass, and
\begin{equation}
  \label{eq:pheno_cuts}
  {p_{\tr\gamma_1}^\MIN} = 25~{\rm GeV},
  %
  \qquad\qquad
  %
  {p_{\tr\gamma_2}^\MIN} = 22~{\rm GeV},
  %
  \qquad\qquad
  %
  {m_{\gamma\gamma}^\MIN} = 25~{\rm GeV}.
\end{equation}
The values of the barred quantities in eq.~(\ref{eq:supp_fact_Born_MiNNLO})
and of the power $a$ have been set to
\begin{equation}
\bar{p}_{\tr \gamma} = 22~{\rm GeV},\qquad\qquad  \bar{R}_{j \gamma} = 0.4\,,
\qquad\qquad a = 1\,.
\end{equation}
In addition, the {\sc\small Powheg Box Res}{} code has been run with the flag {\tt doublefsr}
set to 1. This parameter was first introduced in ref.~\cite{Nason:2013uba},
and we refer the interested reader to this paper for further details.

\subsection{Validation for Drell-Yan and Higgs boson production}
\label{sec:DY_and_Higgs}

In this section, we compare the results obtained with the
original {\sc\small MiNNLO\textsubscript{PS}}{} method
used
in ref.~\cite{Monni:2020nks} for Drell-Yan and Higgs boson production against
those obtained with the new formulation spelled in
section~\ref{sec:MiNNLO}. As discussed in that section,
one expects the new formulation to yield results compatible with the original
method for processes where the size of non-singular corrections is small compared
to the total cross section.

An estimate of the size of the non-singular correction can be obtained by comparing
the total NNLO cross section against the integral of the first term on the
right-hand side of eq.~(\ref{eq:dsig_final}) over the full phase space.  The
results obtained for diphoton, Drell-Yan and Higgs boson production are the
following
\begin{equation}
  \begin{array}{ll}
    \sigma_{\sss\rm NNLO}^{\gamma\gamma} = 155.7 \pm 1.0~{\rm pb}\,,\qquad \qquad  &
    \sigma_{r}^{\gamma\gamma}=55.7 \pm 0.6~{\rm pb}\,,
    \\[2mm]
    \sigma_{\sss\rm NNLO}^{\rm\sss DY}=1919\pm 1~{\rm pb}\,,  &
    \sigma_{r}^{\rm \sss DY} = 1904\pm 3~{\rm pb}\,,
    \\[2mm]
    \sigma_{\sss\rm NNLO}^{\rm\sss H}= 39.64 \pm 0.01~{\rm pb}\,,  &
    \sigma_{r}^{\rm \sss H} = 34.03 \pm 0.07~{\rm pb}\,.
  \end{array}
\end{equation}
For diphoton production, $\sigma_{r}^{\gamma\gamma}$ contributes to only about
one third of the total cross section, at difference with Drell-Yan and Higgs
boson production, thereby justifying the choices made in this paper.

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/validation-y_DY.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/validation-y_H.pdf}
  \end{center}
  \caption{Comparison between fixed-order predictions from \Matrix{}
    against the two {\sc\small MiNNLO\textsubscript{PS}}{} results (at ``Les Houches'' level)
    obtained using the original formulation of the method (labelled as
    {\tt no\,FOatQ}) and the new one presented in this
    work (labelled as {\tt FOatQ}). In the left pane we show the
    rapidity distribution of the dilepton system in Drell-Yan production,
    whereas in the right pane we show the Higgs boson rapidity.
  The ratios with respect to the \Matrix{} results are also shown in the lower
  panel. }
  \label{fig:validation}
\end{figure}
As a further validation, in figure~\ref{fig:validation} we compare the
rapidity distribution of the color singlet for Drell-Yan and Higgs boson
production. We show the distributions obtained with the original formulation,
where the finite terms are evaluated at $\mu=p_{\sss T}$ (labelled as {\tt no\,FOatQ}), and
the new one, where such terms are evaluated at $\mu=Q$, with $Q$ the
invariant mass of the color singlet (labelled as {\tt FOatQ}).  The {\sc\small MiNNLO\textsubscript{PS}}{}
results shown in the figure are obtained after generating the {\sc\small Powheg}{}
hardest emission, i.e.~at the ``Les Houches'' level. In the plots we also
show the NNLO results from \Matrix{}, obtained setting $\mu=Q$. For this
comparison only, we use the same PDF sets as those used in
ref.~\cite{Monni:2020nks}.
In the lower inserts of the figure we plot the ratio of the displayed
distributions with respect to the \Matrix{} result.

The curves show a very good agreement between the NNLO and the {\sc\small MiNNLO\textsubscript{PS}}{}
results obtained with either formulations, both for the central scale and the
uncertainty band. In particular, since the Drell-Yan process features a very
small perturbative uncertainty band, the remarkable agreement between the
NNLO and {\tt FOatQ}{} curves displayed in the left pane of
figure~\ref{fig:validation} is a robust validation of the new formulation of
the {\sc\small MiNNLO\textsubscript{PS}}{} method.

\subsection{Diphoton production: comparison with \Matrix}
\label{sec:comparison_with_matrix}

In this section we validate the differential cross section of eq.~(\ref{eq:dsig_final})
against the fixed-order NNLO one implemented
within the public version of the \Matrix{} code.
We expect the two results to agree up to terms beyond the NNLO accuracy.
The \Matrix{} results have
been obtained setting the slicing parameter $r_{\sss\rm cut} =
0.0005$ and for a central scale choice $\mur=\muf=Q$. Using the isolation
criteria and the fiducial cuts reported in
section~\ref{sec:physical_parameters}, the total cross section computed by
\Matrix{} and {\sc\small MiNNLO\textsubscript{PS}}{} are in agreement within the statistical errors, and
they read respectively
\begin{equation}
  \sigma_{\rm\sss tot}^{\rm \sss Matrix} = 155.7 \pm 1.0~{\rm pb}\,,
  %
  \qquad\qquad
  %
  \sigma_{\rm\sss tot}^{\rm \sss MiNNLO} = 154.9 \pm 0.2~{\rm pb}\,.
\end{equation}

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-m_gamgam.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-y_gamgam.pdf}
  \end{center}
  \caption{Comparison between fixed-order results from \Matrix{} and
    {\sc\small MiNNLO\textsubscript{PS}}{} for the diphoton invariant mass and rapidity, for the default
     central value of the renormalization and factorization scales. The
    statistical errors are also shown.}
  \label{fig:MiNNLO_NNLO_aa}
\end{figure}
In figure~\ref{fig:MiNNLO_NNLO_aa} we plot the distributions for the
invariant mass and rapidity of the diphoton system computed with \Matrix{}
and {\sc\small MiNNLO\textsubscript{PS}}{}, in the top panes, and the ratio of the two curves in the
lower ones. The figure shows a reasonably good agreement between the two
curves within the statistical errors.  We ascribe the mild difference in the
high invariant-mass region to effects beyond the nominal accuracy of our
result, which can differ from the purely fixed-order \Matrix{} one by
higher-order effects, present in eq.~(\ref{eq:dsig_final}).
We have explicitly verified that, by applying a transverse momentum cut on
the diphoton system, the same trend is also present when we compare the exact
fixed-order NLO result for $\aaj$ production against the {\sc\small MiNNLO\textsubscript{PS}}{} one:
when we move away from the region of small transverse momentum of the
diphoton system that is affected by large logarithms, we still observe a mild
difference between the central values of the two distributions, but still within
the scale-variation bands.

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-m_gamgam-band.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-y_gamgam-band.pdf}
  \end{center}
  \caption{Comparison between fixed-order results from \Matrix{} and
    {\sc\small MiNNLO\textsubscript{PS}}{} for the diphoton invariant mass and rapidity, as in
    figure~\ref{fig:MiNNLO_NNLO_aa}. Uncertainty bands from scale
    variations are also shown.}
  \label{fig:MiNNLO_NNLO_aa_band}
\end{figure}

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-pT_gam1-band.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/MiNNLO_NNLO-pT_gam2-band.pdf}
  \end{center}
  \caption{Comparison between fixed-order results from \Matrix{} and
    {\sc\small MiNNLO\textsubscript{PS}}{} for the transverse momentum of the hardest and
    next-to-hardest photon. Uncertainty bands from scale
    variations are also shown.}
  \label{fig:MiNNLO_NNLO_pta_band}
\end{figure}

The theoretical uncertainties on the \Matrix{} and {\sc\small MiNNLO\textsubscript{PS}}{} total cross
sections, estimated through a seven-point scale variation obtained by
multiplying and dividing the central renormalization and factorization scales by a
factor of 2, are given by\footnote{\Matrix{} provides also extrapolated
values for the total cross section down to $r_{\sss\rm cut} = 0$, that, for
the central scale, is equal to $\sigma_{\rm\sss tot}^{\rm \Matrix} = 153.9
\pm 1.9$~pb.  We notice that, with the extrapolation process, the associated
statistical error is larger than in the non-extrapolated one. For comparison
with eq.~(\ref{eq:scal_var_XS}), the extrapolated scale variation band
is $\pm 4$\%.}
\begin{equation}
  \label{eq:scal_var_XS}
  \sigma_{\rm\sss tot}^{\rm \sss \Matrix} = 155.7\, ^{+5\%}
  _{-4\%}~{\rm pb}
  %
  \qquad\qquad
  %
  \sigma_{\rm\sss tot}^{\rm \sss MiNNLO} = 154.9 \, ^{+6\%}
  _{-5\%}~{\rm pb}\,,
\end{equation}
and are in good agreement.
In figures~\ref{fig:MiNNLO_NNLO_aa_band} and~\ref{fig:MiNNLO_NNLO_pta_band}
we compare the scale variation bands obtained from the two codes for the
invariant mass and rapidity distributions of the diphoton system and the
transverse momentum of the hardest and next-to-hardest photon.
We find an overall good agreement among the different curves, and compatible
size for the scale-variation bands.

\section{Phenomenological results}
\label{sec:results}

After validating the implementation of the {\sc\small MiNNLO\textsubscript{PS}}{} differential cross
section for diphoton production given in eq.~(\ref{eq:dsig_final}), in this
section we present some distributions of physical interest obtained from the
generated events, before and after passing them through a parton shower.  We
have generated about 16 million events without any generation cut, except for
a minimum invariant mass of the diphoton system of 10~GeV.\footnote{We point
out that such generation cut has no effect on the final kinematic
distributions if the fiducial cut on the diphoton invariant mass at the
analysis stage is greater than it.}
As stressed previously, except for this last constraint on the invariant
mass, no other cuts have been imposed, so that these events can be used to
make predictions with arbitrary fiducial cuts.

\subsection{Results at partonic level}
\label{sec:partonic}
In this section we compare the \Matrix{} results against those obtained after
generating the {\sc\small Powheg}{} hardest emission of
eq.~(\ref{eq:MiNNLOPS+Powheg_formula}), often denoted as ``Les Houches
events''~(LHE), and labelled in the figures as ``{\sc\small MiNNLO\textsubscript{PS}}~(LHE)''.
\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/NNLO_LH-m_gamgam.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/NNLO_LH-y_gamgam.pdf}
  \end{center}
  \caption{Comparison between the distributions obtained from the
    fixed-order calculation from \Matrix{} and the results after the
    generation of the hardest emission, for the diphoton invariant
    mass and rapidity.}
  \label{fig:NNLO_LH_aa}
\end{figure}

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/NNLO_LH-pT_gam1.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/NNLO_LH-pT_gam2.pdf}
  \end{center}
  \caption{Comparison between the distributions obtained from the
    fixed-order calculation from \Matrix{} and the results after the
    generation of the hardest emission, for the transverse momentum of
    the hardest and next-to-hardest photon.}
  \label{fig:NNLO_LH_pta}
\end{figure}
In figure~\ref{fig:NNLO_LH_aa} we show results for the invariant mass and
the rapidity of the diphoton system, whereas in
figure~\ref{fig:NNLO_LH_pta} we display the transverse momentum of the
hardest and next-to-hardest photon.
We find  good agreement between the \Matrix{} and the
{\sc\small MiNNLO\textsubscript{PS}}{}~(LHE) predictions, both for the central value and for the
uncertainty bands due to scale variations. Notice that, at variance
with similar comparisons for other processes not involving photons
(like Drell-Yan or massive diboson production), this is not trivial
due to the presence of an isolation criterion  in the
definition of the process.

\subsection{Results after parton showering}
\label{sec:shower}
In this section we show results obtained after the completion of the parton
shower performed by \PythiaEight{}~\cite{Sjostrand:2014zea,
  Sjostrand:2006za}. We rely on the \PythiaEight{} interface to the
{\sc\small Powheg Box Res}{}, provided by the {\tt main31} configuration file, distributed
with \PythiaEight{}.  The results presented in this section are obtained
after switching off multi-parton interactions, QED radiation and
hadronization effects, using the Monash tune~\cite{Skands:2014pea}.
We set the \PythiaEight{} parameter {\tt POWHEG:pThard} to 2~(i.e.~we use the
prescription introduced in section~4 of ref.~\cite{Nason:2013uba}), and the
{\tt SpaceShower:dipoleRecoil} to 1.

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/LH_PY8-m_gamgam.pdf}\ \
    \includegraphics[width=0.49\textwidth]{figures/LH_PY8-y_gamgam.pdf}
  \end{center}
  \caption{Comparison between the distributions obtained from the
    {\sc\small Powheg}{} events before and after the \PythiaEight{} parton shower
    for the diphoton virtuality and rapidity.}
  \label{fig:LH_PY8_aa}
\end{figure}

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{figures/LH_PY8-pT_gam1.pdf}
    \includegraphics[width=0.49\textwidth]{figures/LH_PY8-pT_gam2.pdf}
  \end{center}
  \caption{Comparison between the distributions obtained from the
    {\sc\small Powheg}{} events before and after the \PythiaEight{} parton shower
    for the transverse momentum of the hardest and next-to-hardest
    photon.}
  \label{fig:LH_PY8_pta}
\end{figure}
In figures~\ref{fig:LH_PY8_aa} and~\ref{fig:LH_PY8_pta}, we present the same
distributions of the previous sections, but after the \PythiaEight{} shower.
We observe a reduction of around 5--10\% of the differential cross sections,
compared to the {\sc\small MiNNLO\textsubscript{PS}}~(LHE), with the largest values in the region of
high transverse momenta of the two photons.  We ascribe this behaviour to the
fact that, with the increased multiplicity of the partonic/hadronic activity
due to the \PythiaEight{} shower, the photons in the showered events are less
likely to satisfy the isolation criteria, giving a smaller cross section.  We
observed such pattern also for similar distributions computed with a
completely independent code for diphoton production, built with the standard
{\sc\small Powheg}{} NLO+PS formalism, reassuring us that the above interpretation is
correct, and that this behaviour is not due to some {\sc\small MiNNLO\textsubscript{PS}}{} features.
We leave further investigation on this aspect for future studies.

\subsection{Comparison with ATLAS results}
\label{sec:atlas}

\def\ensuremath{{p_{\sss T}}_{\gamma\gamma}}\xspace{\ensuremath{p_{\perp,\gamma\gamma}}\xspace}
\def\ensuremath{{p_{\sss T}}_{\gamma\gamma}}\xspace{\ensuremath{{p_{\sss T}}_{\gamma\gamma}}\xspace}
\def\ensuremath{\phi_{\eta}^*}\xspace{\ensuremath{\phi_{\eta}^*}\xspace}
\def\ensuremath{\pi - \Delta\phi_{\gamma\gamma}}\xspace{\ensuremath{\pi - \Delta\phi_{\gamma\gamma}}\xspace}
\def\ensuremath{a_{\perp\gamma\gamma}}\xspace{\ensuremath{a_{\perp\gamma\gamma}}\xspace}

In this section we compare our results with those presented by the ATLAS
Collaboration in ref.~\cite{ATLAS:2021mbt} obtained at 13~TeV.
In order to make a fair comparison with the experimental data, we have
added to the {\sc\small MiNNLO\textsubscript{PS}}{} diphoton contribution, presented in this
paper, the leading-order contribution coming from the gluon-induced
production of a photon pair through a closed quark loop (denoted by
$gg\rightarrow\gamma\gamma$ in the following), which is of the same order of
the NNLO corrections and is further enhanced by the particularly
sizable gluon luminosity.  As we consider this contribution only at
leading order, we simulate it at LO+PS accuracy, setting the
renormalization and factorization scales, and the upper limit for the
transverse momentum of the subsequent shower evolution, equal to the
invariant mass of the diphoton system.  The analytic amplitudes for
this process have been taken from ref.~\cite{Bern:2002jx} and
implemented in the \RES{} framework, neglecting the top-quark loop
contribution~\cite{Dicus:1987fk}.
This contribution amounts for at most a few percent in certain regions of the
kinematic distributions we are plotting. Its inclusion would improve the agreement
with data.

The analysis of our events is performed using
\rivet~\cite{Bierlich:2019rhm}, with the same \PythiaEight{} settings
as those in the previous section. The fiducial volume is defined by
the following requirements
\begin{equation*}
  {p_{\tr\gamma_1}} > 40~{\rm GeV},
  %
  \qquad
  %
  {p_{\tr\gamma_2}} > 30~{\rm GeV},
  %
  \qquad
  %
  \Delta R_{\gamma\gamma} > 0.4\,,
\end{equation*}
\begin{equation}
  \label{eq:atlas_cuts}
  |y_\gamma| < 1.37\,, \qquad 1.52 < |y_\gamma| < 2.37.

\end{equation}
We do not report here the exact photon-isolation criteria which are
described in detail in section~4.1 of ref.~\cite{ATLAS:2021mbt}.

In figures~\ref{fig:ATLAS_yy_m_pt} and~\ref{fig:ATLAS_y_pts}
\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/yy_m-full.pdf}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/yy_pT-full.pdf}
  \end{center}
  \caption{Comparison between the ATLAS data and the distributions obtained
    with {\sc\small MiNNLO\textsubscript{PS}}{}+\PythiaEight{}, for the diphoton invariant mass and its
    transverse momentum. Scale-variation bands are also shown, together with
    the statistical errors of the central value.}
  \label{fig:ATLAS_yy_m_pt}
\end{figure}
\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/ph1_pt-full.pdf}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/ph2_pt-full.pdf}
  \end{center}
  \caption{Comparison between the ATLAS data and the distributions obtained
    with {\sc\small MiNNLO\textsubscript{PS}}{}+\PythiaEight{}, for the transverse momentum of the
    leading and subleading photon. Scale-variation bands are also shown,
    together with the statistical errors of the central value.}
  \label{fig:ATLAS_y_pts}
\end{figure}
we show the invariant mass and the transverse momentum of the diphoton
system, and the transverse momentum of the leading and subleading
photon, respectively. Overall we find a reasonably good agreement
between data and theoretical predictions.

For the invariant mass distribution $m_{\gamma\gamma}$ we observe a good
agreement in the bulk of the cross section. Given the cuts of
eq.~(\ref{eq:atlas_cuts}), the region where $m_{\gamma\gamma}< 80$~GeV
is populated only by $\gamma\gamma$ + jet(s) events, therefore
our results are only NLO+PS accurate, as confirmed also by the wider
uncertainty bands. For $m_{\gamma\gamma}\lesssim 40$~GeV, {\sc\small MiNNLO\textsubscript{PS}}{}
results overshoot ATLAS data by an amount compatible with what has been
observed, for other predictions of similar accuracy, in
ref.~\cite{ATLAS:2021mbt}. This is particularly true for the first bin.
The fact that this region is characterized
by a large NLO $K$-factor~\cite{Catani:2018krb} hints at the possibility
that the inclusion of higher-order corrections will improve the
agreement with data.  At large $m_{\gamma\gamma}$ values we observe
differences up to about 15\%. Such differences might be due to the
absence of higher-order effects. Top-quark mass effects
above the threshold $m_{\gamma\gamma}\simeq 2 m_t$, that we are
neglecting in the quark-induced 2-loop amplitudes as well as in the
$gg\rightarrow\gamma\gamma$ channel, can also induce differences at a
few percent level.

As far as the transverse-momentum spectrum of the diphoton pair is concerned,
our predictions are compatible with experimental data across most of the
$\ptAA$ spectrum, within the estimated theoretical uncertainties.
Due to the all-order treatment of soft and collinear radiation encoded in the
{\sc\small MiNNLO\textsubscript{PS}}{} Sudakov form factor and in the subsequent matching to {\sc\small Pythia}{},
the region of small $\ptAA$ is in a fair agreement with the data, within the
statistical error and the scale-variation band.
For large values of $\ptAA$, data and
theoretical predictions agree well in shape but display a flat offset of
about 10\%.

The comparison of our results against ATLAS data for the $p_{\sss T}$ spectrum of
the leading and subleading photon is shown in
figure~\ref{fig:ATLAS_y_pts}. Data and theoretical predictions agree quite
well over the full range of the transverse momentum.

We would also like to point out that the choices for the central value of the
renormalization and factorization scales, in a fixed-order computation, can
have a sizable impact on the theory predictions at large invariant mass and
transverse momentum of the diphoton system, as discussed in detail e.g.~in
ref.~\cite{Gehrmann:2020oec}. The size of these effects are a consequence of
a slowly convergent perturbative series.
Assessing this type of effects by exploring different scale choices in our
{\sc\small MiNNLO\textsubscript{PS}}{} simulation goes beyond the scope of this paper, although this is
a-priori possible, both for the handling of the large-$p_{\sss T}$ limit of the
singular component (first term of eq.~(\ref{eq:dsig_final})), as discussed
for instance in ref.~\cite{Mazzitelli:2021mmm}, as well as for the scales
used in the non-singular contribution (last term in the same equation).

\begin{figure}[htb!]
  \begin{center}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/yy_piMDphi-full.pdf}
    \includegraphics[width=0.49\textwidth]{ATLAS_plots/yy_phiStar-full.pdf}
  \end{center}
  \caption{Comparison between the ATLAS data and the distributions obtained
    with {\sc\small MiNNLO\textsubscript{PS}}{}+\PythiaEight{}, for the azimuthal separation of the two
    photons $\Delta\phi_{\gamma\gamma}$ and for \ensuremath{\phi_{\eta}^*}\xspace, as defined in
    eq.~(\ref{eq:phistar}).  Scale-variation bands are also shown, together
    with the statistical errors of the central value.}
  \label{fig:ATLAS_yy_phi}
\end{figure}

In figure~\ref{fig:ATLAS_yy_phi} we show the differential cross section as a
function of two angular observables: the azimuthal separation
$\Delta\phi_{\gamma\gamma}$ of the two photons and \ensuremath{\phi_{\eta}^*}\xspace, defined as
\begin{equation}
  \label{eq:phistar}
  \ensuremath{\phi_{\eta}^*}\xspace=\tan\frac{\ensuremath{\pi - \Delta\phi_{\gamma\gamma}}\xspace}{2}\,\sin\theta_{\eta}^* = \tan\frac{\ensuremath{\pi - \Delta\phi_{\gamma\gamma}}\xspace}{2}\,
  \sqrt{1-\left(\tanh\frac{\Delta y_{\gamma\gamma}}{2}\right)^2},
\end{equation}
where $\Delta y_{\gamma\gamma}$ is the azimuthal angular separation of the
two photons. Such variable, first introduced for Drell-Yan processes in
ref.~\cite{Banfi:2010cf}, is known to be sensitive to the same dynamics
governing the $\ensuremath{{p_{\sss T}}_{\gamma\gamma}}\xspace$ spectrum, but it allows for a better resolution at
small values of $\ensuremath{{p_{\sss T}}_{\gamma\gamma}}\xspace$.  The agreement between data and the theoretical
predictions is rather good on the whole range within the statistical and
scale-variation bands. We stress that, for $\Delta \phi_{\gamma\gamma} \rightarrow
\pi$, the NNLO fixed-order results for the above distributions are divergent,
while we get a finite distributions due to the {\sc\small MiNNLO} Sudakov
form factor.

\section{Conclusions}
\label{sec:conclusions}

In this paper we have presented the implementation of a new {\sc\small MiNNLO\textsubscript{PS}}{} event
generator accurate at NNLO in QCD for the production of a photon pair in
hadronic collisions, within the {\sc\small Powheg Box Res}{} framework.
This implementation is based on the {\sc\small MiNNLO\textsubscript{PS}}{} formalism presented in
refs.~\cite{Monni:2019whf, Monni:2020nks}, that we have modified in order to
deal with the new features needed to describe a process with photons at the
Born level.
We have devised and implemented a general method to deal with the presence of
QED singularities at the Born level within the {\sc\small Powheg Box Res}{}, and we have
presented a new mapping from $\gamma\gamma j$ to the $\gamma\gamma$.
The combined use of the new mapping and of the method to deal with the QED
divergences allowed us to simulate diphoton production at NNLO+PS accuracy
without introducing any generation or technical cut.
Our generator produces weighted events that cover the entire phase space,
thereby allowing the generated sample to be used with arbitrary fiducial
cuts.

Furthermore, we have introduced a few modifications to the {\sc\small MiNNLO\textsubscript{PS}}{}
formalism, that do not change the formal accuracy of the method and reduce
the numerical impact of higher-order terms, thus leading to better agreement
with fixed-order computations.  We have also verified that the new
formulation of {\sc\small MiNNLO\textsubscript{PS}}{}, when applied to the original {\sc\small MiNNLO\textsubscript{PS}}{}
implementations with massive bosons, gives results fully compatible with the
original one.

Finally, we have presented some distributions of physical interest obtained
from the generated events both before and after passing them through the
parton shower,
and compared them with the ATLAS
results at 13~TeV. Using the standard settings of \PythiaEight{} when
interfacing it with the {\sc\small Powheg Box Res}{}, we observe a reduction of about 5–10\%
of the inclusive differential cross sections compared to the
NNLO fixed-order ones.  We ascribe this effect to the increased
multiplicity of the partonic/hadronic activity coming from the shower causing
the photons to less likely satisfy the isolation criteria.  Further
investigation of the matching procedure between
{\sc\small Powheg Box Res}{}+{\sc\small MiNNLO\textsubscript{PS}}{} and \PythiaEight{} is left for future studies.

\acknowledgments
We would like to thank S.~Alioli, A.~Broggio and J.~Lindert for useful
discussions.
We also thank J.P.~Guillet, P.F.~Monni, and M.~Wiesemann
for carefully reading the manuscript and for their useful comments.

E.R. thanks P.F.~Monni and M.~Wiesemann for several discussions about
theoretical aspects related to the {\sc\small MiNNLO\textsubscript{PS}}{} method. In particular, the
relevant aspects of the formulation of the {\sc\small MiNNLO\textsubscript{PS}}{} method with the scale
choice discussed in section~\ref{sec:MiNNLO}, and their initial
implementation in the code, were developed in collaboration with them.

The work of A.G. was supported by the ERC Starting Grant REINVENT-714788.

redpajama_set_name: RedPajamaArXiv


text:
Hip in Detroit's Picks for the Weekend: September 27-29
Photo Credit: Liam MacRae
Friday, September 27 - Kanye West's Sunday Service @ Aretha Franklin Ampitheatre: In case you haven't heard, Kanye West is bringing his Sunday Service to Detroit this Friday. The event was first mentioned when auditions for singers and choirs were announced online. Then, on Thursday, the details were finalized and the frenzy for grabbing free tickets via Ticketmaster was on. Unfortunately those free tickets went fast. So, the show is this Friday at Aretha Franklin Ampitheatre and it is free, but if you didn't get those free tickets via Ticketmaster, you are S.O.L. Some tickets are being sold on the Facebook event here, but they're going for about $150 and they most likely aren't legit. Good luck getting into this one!
Friday, September 27 - Eric Andre @ The Fillmore: Comedian Eric Andre brings his Legalize Everything tour to town this Friday night. He will be performing live at The Fillmore at 8 p.m. with doors opening at 7 p.m. Tickets start at $35 and somehow, there are still some available. You can purchase yours online here.
Saturday, September 28 - Founders Detroit: Outer Limits Party 2019: Some of our absolute favorites are playing together at Founders this Saturday! The brewery will be holding an outdoor party with live music from Jyarsch, ERNO, Handgrenades, and Jessica Hernandez & The Deltas! This is your last chance to see Jess before she heads back to LA, so don't miss it! Tickets are $20 and and can be purchased in advance here. 21+ are welcome. The party goes from noon until midnight and the music begins at 2 p.m.
Saturday, September 28 - Bowlero Grand Opening Party: A few weeks ago, we told you about the hippest new bowling alley in town, Bowlero. Well, this Saturday night, they are holding their official grand opening party! Doors open at 11 a.m. with DJs all day. At night, there will also be live music in the lounge with Party!Zeppelin kicking things off at 7 p.m., followed by The High Strung, CATL, and Friends Of Dennis Wilson. The fun goes until 2 a.m., cover is free, and 18+ are welcome in the lounge, while all ages are welcome in the bowling alley.
Sunday, September 29 - PinzuCon3 @ The Magic Stick: PinzuCon is back and this time around, it has a new home! PinzuCon will be holding their third convention this Sunday afternoon at The Magic Stick in Detroit. The event, which was previously held at The Rust Belt Market, brings together artists and vendors selling all sorts of pins! For this installment of the event, they will be highlighting Halloween/Horror themed makers, but there will be plenty of other pins too. All ages are welcome, cover is free, and the event runs from 11 a.m. 'til 7 p.m.
Labels: Aretha Franklin Ampitheatre, Bowlero Lanes & Lounge, Eric Andre, Erno, Founders, Handgrenades, Jessica Hernandez and The Deltas, Kanye West, PinzuCon, Sunday Service, The Fillmore, The Magic Stick
Win Tickets to See Radkey at Small's
Win Tickets to See The Joy Formidable and DeVotchK...
Celebrate Small's 20 Year Anniversary this Weekend...
Hip in Detroit's Picks for the Weekend: September ...
Win Tickets to See Sum 41 at The Fillmore
24 Hour Ticket Giveaway: KOKOKO! at Deluxx Fluxx
Vans Gives Back To Detroit
Tunde Olaniran and The Mosaic Youth Theatre is Bri...
The Empire Strikes Back in Concert
24 Hour Giveaway: Nightmares on Wax @ Deluxx Fluxx...
Eastern Market Comes Alive After Dark this Thursda...
It's Time to Check Out Coup D'état
DIY Street Fair Takes Over Ferndale this Weekend
Win Tickets to See The HU at El Club
Win Tickets to See Angels & Airwaves at The Fillmo...
Win Tickets to See K.Flay at The Crofoot
The SheTown Film Festival Returns to Detroit this ...
Win Tickets to Michiganders Album Release Party at...
Win Tickets to See Big Business at Deluxx Fluxx
It's Time to Dally in The Alley

redpajama_set_name: RedPajamaCommonCrawl


This Bridal Pearls in Swarovski Brown Pearls with silver rondells spacer with drop down accent your bridal gown & the bridesmaids dresses too. We used Swarovski Brown Pearls are top quality & we offer many items in Swarovski Pearls. Bridal jewelry & bridesmaids jewelry are great gifts & our wedding Pearls jewelry is classic & elegant. Dont forget the mothers when ordering Pearls wedding jewelry. Gorgeous Drop down Necklace diamante silver rondells spacer Necklace with Swarovski Brown Pearls Set. FashionJewelryForEveryone & Gifts is well known for our superb collection of personalized bridesmiad jewelry & bridal jewelry that will last a lifetime. Select from our large collection of bridesmaids gift ideas for your wedding party.


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Q: Pynput Moving the Mouse in Minecraft I'm trying to create a macro in Minecraft and I need to move the mouse, but in-game, mouse.move(-100, 0) won't move the mouse. Outside of Minecraft I can move the mouse, but in-game I can't. I'm on macOS Big Sur and I don't think I have any more relevant info. Thanks in advance.

A: I'm in a similar situation. Here's another link that I just found and plan to use:
How to control the mouse in Minecraft using Python?

redpajama_set_name: RedPajamaStackExchange


The Backyard Garden Club has been sizing up local residential gardens since 1986, but now they've decided to showcase some of Dundee Township's loveliest yards.
"A Walk Through Our Garden Gates" 10 a.m. to 4 p.m. Saturday will open four gardens each in West Dundee and Sleepy Hollow. Door prizes and a raffle will also be offered.
Tickets may be purchased for $9 the day of the walk at 831 Winmoor Drive in Sleepy Hollow. Proceeds will fund a scholarship for a college student in horticulture. For information, call 847-695-7540.


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Person-centric Medicaid: Gingrich, Mehrotra account for factors to help coordinate Medicaid care
Medicaid has to evolve into a delivery model that takes into account the uniqueness of each individual-both their specific health status, and issues outside of the traditional healthcare system, such as transportation, living conditions and substance abuse problems, according to insight from Newt Gingrich, founder of the Center for Health Transformation (CHT) and Rishabh Mehrotra, president and CEO of SHPS, a provider of health advocacy and health benefits solutions.
"In order to drive value in the Medicaid system, we need to create a person-centric approach to healthcare. This means we need to understand health risk factors and coordinate care across all services supporting individual well-being," says Mehrotra. "Large employers are adopting person-centric healthcare strategies, including creating coalitions, integrating health advocacy with incentives and adopting innovative plan designs. As a result, they're seeing improved employee health and cost savings of 20%. There's no reason Medicaid can't put the same approaches in place."
Former House Speaker Gingrich tells MANAGED HEALTHCARE EXECUTIVE that the bias in healthcare for the 20th century was profession-centered rather than individual-centered and was primarily focused on acute care.
"In addition, we have adopted a wage and price control system, whereby businesses have been encouraged to pick up the cost of healthcare," Gingrich continues. "We created a system where people have had no personal investment in the cost of healthcare. As a result of that, there's no natural center of people asking about value and choice."
In Making Medicaid Work: A Practical Guide for Transforming Medicaid, co-authored by CHT and SHPS, Mehrotra and Gingrich propose a series of principles for understanding how Medicaid must look in the future if it is to be a higher quality, fiscally sustainable program. They propose four key recommendations:
Q A viable electronic health record (EHR) system is still years away-how can we create a better system when health IT is still just emerging?
Gingrich: It is true that universal adoption of EHRs may still be years away, but information technology has advanced far enough that states and other entities can begin reaping benefits of an EHR system today.
Mehrotra: Some of the more advanced care management vendors now have the IT infrastructure to compile a 360-degree view of a member including their complete claims history, their care plan, healthcare providers and health status in one location. This holistic snapshot of the individual provides significant value and utility for the provider, who could access it with the patient's permission. By providing a comprehensive view of a member's health claims, clinical risks and care plan, we're making rapid and immediate improvement from where the industry has been to date.


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Q: mysql foreign key as primary key error I am facing foreign key error in my SQL. My mySQL tables are
create table if not exists table1(
cust_name varchar(30) not null,
cust_phone char(16) not null,
cust_mail char(30) not null,
cust_address varchar(100),
bills int(10),
primary key (cust_name)
)ENGINE=INNODB;

create table if not exists table2 (
camp_id varchar(30) not null,
advr_id varchar(30) not null,
foreign key (advr_id) references table1 (cust_name),
primary key (camp_id)
)ENGINE=INNODB;

create table if not exists table3(
ad_id varchar(30) not null,
camp_id varchar(30) not null,
foreign key (camp_id) references table2(camp_id),
primary key (ad_id)
)ENGINE=INNODB;

create table if not exists counter(
ad_id varchar(30) not null,
foreign key (ad_id) references table3(ad_id),
PRIMARY KEY(ad_id)
)ENGINE=INNODB;

When I run above commands, I get follwoing error.

ERROR 1215 (HY000) at line 96: Cannot add foreign key constraint

What is the problem?

A: There is no any error. This is run without error.
mysql> create table if not exists table1(
    -> cust_name varchar(30) not null,
    -> cust_phone char(16) not null,
    -> cust_mail char(30) not null,
    -> cust_address varchar(100),
    -> bills int(10),
    -> primary key (cust_name)
    -> )ENGINE=INNODB;
Query OK, 0 rows affected, 1 warning (0.08 sec)

mysql> create table if not exists table2 (
    -> camp_id varchar(30) not null,
    -> advr_id varchar(30) not null,
    -> foreign key (advr_id) references table1 (cust_name),
    -> primary key (camp_id)
    -> )ENGINE=INNODB;
Query OK, 0 rows affected, 1 warning (0.10 sec)

mysql> create table if not exists table3(
    -> ad_id varchar(30) not null,
    -> camp_id varchar(30) not null,
    -> foreign key (camp_id) references table2(camp_id),
    -> primary key (ad_id)
    -> )ENGINE=INNODB;
Query OK, 0 rows affected (0.37 sec)

mysql> create table if not exists counter(
    -> ad_id varchar(30) not null,
    -> foreign key (ad_id) references table3(ad_id),
    -> PRIMARY KEY(ad_id)
    -> )ENGINE=INNODB;
Query OK, 0 rows affected (0.42 sec)

I Execute in my database but there is no any error or warning

A: There is no error when I run this code. I think their might be some spelling mistakes in your origional code. Try to use mysql gui application for details of logs. It will precisely point the error.


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Very good service...
10 Patricia Gonzalez
24 ,48 or 72 hours
The Budapest Hop-On Hop-Off Sightseeing tour is the most practical and comfortable way to explore the Pearl of the Danube. Explore the city at your own pace by hopping off and back on as many times as you wish.
Budapest Hop-On Hop-Off bus
Nighttime tour of Budapest
St Stephen's Basilica
Fisherman's Bastion
Lines and stops
Budapest's sightseeing bus tour offers visitors several routes with over 20 stops. Explore the city's most beautiful landmarks and neighbourhoods on board an open-top double-decker bus. While you admire the panoramic views, learn about the capital's history and monuments with your individual headset with pre-recorded commentary in English or 22 other languages.
Some of the most popular stops include St Stephen's Basilica, Heroes' Square, Margaret Island, Széchenyi Thermal Bath and the Hungarian Parliament.
Check out Big Bus Tours complete red and blue routes and stops on the following Budapest map:
Map of the route and list of stops
Types of tickets
Big Bus offers various types of tickets to best suit your preferences:
1-day ticket: Includes unlimited access to the Big Bus Sightseeing buses for 1 day.
2-day ticket: Offers unlimited access to the Big Bus Sightseeing buses for 2 days. This ticket also covers a cruise down the Danube River. Embark on dock 11 to enjoy impressive views of the Hungarian Parliament Building, Chain Bridge and Buda Castle.
3-day ticket: Offers limitless journeys on the Big Bus Sightseeing tour for 3 days. It also includes a free cruise along the Danube River, a nighttime tour on an open-top double-decker bus, and a free bus ride to and from the Buda Castle.
Nighttime tour: It only includes a nighttime panoramic tour on a open-top bus to admire the city's lit up monuments. The route doesn't include any stops.
Tickets are valid for 1, 2 or 3 consecutive days from the first time you use it. For example, if you purchase a 2-day ticket and activate it on Monday at 3pm, it'll be valid until Wednesday at 3pm.
During its validity, you can hop on and off as many times as you wish.
Schedule and frequency
Red line: 9 am – 4 pm, buses run every 10 minutes.
Blue line: 9:30 am to 6:00 pm, buses run every 10-15 minutes.
Nighttime tour: One departure time at 6pm.
River cruise: every two hours between 1 pm and 9 pm.
Shuttle bus to Buda Castle: from 9:30 am to 4:30 pm.
Children less than 5 years old
1-day ticket
US$ 24.7 US$ 22.2
24 ,48 or 72 hours.
The tour includes audio commentary in English.
Unlimited use of the Big Bus Budapest Hop On Hop Off buses during the chosen period.
Individual headset with interesting commentary in English or 22 other languages.
Cruise down the Danube River (included in the 2 or 3 day ticket).
Nighttime bus tour (included in the 3 day ticket).
Shuttle service to the Castle District (included in the 3 day ticket).
Free WiFi on board.
10% discount on the official ticket price.
Yes (most of the buses are accessible).
from US$ 15.80
Type of activity 1-day ticket 2-day ticket 3-day ticket Night Tour
Non refundable. You may not cancel this activity.
8.0935672506901 171 reviews
5 out of 171 reviews
The on board commentary didn't work on all the seats, the commentary itself had been adapted and obviously pieces had been removed. With the amount of people using the buses plus the money charged, a new commentary each year or season would be appropriate. Also, add the "stop" numbers to it so it ties up with the map - as I said it's been adapted but more likely butchered.
Solihull, United Kingdom
Patricia Gonzalez
San Pedro Garza Garcia, Mexico
Interesting commentary. Saw all the sites. Wish the buses came more frequently.
Jill Deetz
San Diego, United States
La narración apenas indica los edificios nobles junto a los que pasamos, Queda muy pobre
Dalmacio Manuel Rivero Rodriguez
Oviedo, Spain
Budapest Folk Show & Dinner Cruise at 10pm
Spend an unforgettable evening in the capital of Hungary by attending a traditional Hungarian folklore show, followed by a dinner cruise along the elegant Danube River.


While $20 back on a $200 purchase is great, but what if you really having nothing to purchase at Best Buy by the expiration date of September 16, 2012. Well… you can purchase gift cards to then keep in your pocket and use whenever you desire. Best Buy also sells iTunes and Facebook gift cards. I asked about American Express and Visa gift cards and they stated they no longer sell them… bummer!
So this is great for you to pre-buy to use at another time or if you have a gift to give someone. Also, this promotion is valid for purchases online or in store, so if you make the purchase via online, use one of the online shopping portals. If you purchase an iTunes or Facebook gift card, chances are you'll definitely receive the points/cash back, although Best Buy gift cards might not give you the points/cash back as many times they are excluded (although it may work for you!).
@Blue – you are absolutely correct. I am not into selling gift cards, but this absolutely can work!
They sell Amazon cards too!
LX, Brad, and Tracy – thanks so much for letting me know, this is great!
Best Buy in VA also sells Amazon Kindle gift cards, which can be redeemed for anything at Amazon.com.
Do you remember if they sell Lowes or Home Depot gift cards?
@Harold – At my local best buy they did not, however, others have reported different gift cards at their best buy.
Best Buy in Greenville, SC sells Visa gift cards.
Do they sell Amazon gift cards online??
@Nancy – that's great to know! Thanks!
Just picked up a $200 Visa and $200 in Amazon cards while meeting some min spend. Good deal.
@Matt – That's great that you were able to find a location that sells Visa cards!


World Tourism Organization World Tourism Organization law 27228 approving amendments to the statutes. Adopted: 26 November 2015 fact promulgated: 22 December 2015 the Senate and Chamber of deputies of the Argentina nation gathered in Congress, etc. they attest to the force of law: article 1 - approving the modifications of the statutes of the Organization World of the tourism approved the resolutions A/RES/511 (16th) and A/RES/512 (16th), both adopted at the sixteenth meeting of the Assembly General of the World Tourism Organization, held in the city of Dakar - Republic of Senegal-, between November 28 and December 2, 2005 that consist of one (1) item and one (1) Annex respectively, authenticated copies form part of this Act.
-REGISTERED UNDER NO. 27228 - JULIAN A. DOMINGUEZ. -GERARDO ZAMORA. -Lucas Chedrese. -John H. Estrada.


text:
Q: Pip does not work in a virtual environment. Fish-Shell, I use a fish shell, activate a virtual environment, Python works from the environment, but pip is used globally. How to fix it?
$ which python
/develop/venv/bin/python
$ which pip
/usr/bin/pip
$ python -m pip --version
pip 21.1 from /develop/venv/lib/python3.9/site-packages/pip

~ / .pydistutils.cfg The file does not exist.


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My Social Concern Journey
Mission of the Church / By admin
MY JOURNEY WITH SOCIAL CONCERN
In my youth, I belonged to the evangelical minority in a mainline church in Sri Lanka and was dismayed by the lack of emphasis on evangelism in the church. With my commitment to Christ in my mid-teens also came a burden to reach the lost with the gospel. I viewed with suspicion the strong emphasis on social action and justice that characterized the understanding of mission among many in my church almost to the exclusion of emphasis on the need to take the gospel to lost people. In fact the idea that people without Christ were lost was viewed with disdain by many leaders in our church.[1] I had inherited the attitudes of mutual suspicion that existed during the era of the battle for mission that characterized liberal-evangelical relations in the last century.
Yet from my teens I was burdened with a love for the poor and with a sense of a call to serve them. This never left me; and during the past three and a half decades my ministry has been primarily with the urban poor. As a youth I developed a growing unease and anger over the class distinctions and inequality that existed in our society. I knew Christians must address these issues. When I was in my mid-twenties I was exposed to the writings of evangelicals, like John Stott and Ron Sider, which showed me biblical Christians can hold a commitment to evangelism alongside a commitment to social justice.
God called me to work in an evangelistic organization, Youth for Christ. This meant that the primary agenda of our movement was evangelism. But as we began working with the poor we realized that our lifestyle and ministry must take into account the glaring inequalities and injustices that exist in society. According to the laws of our land, those doing evangelism cannot be involved in major social projects because of the perceived possibility of alluring people to Christianity using material incentives. But when one works with the poor one cannot ignore the huge social needs they face. As we are committed to the nurture of those who come to Christ in our ministry we soon found that doing something about their needs was an important part of that nurture. So education and vocational guidance, guidance on how people could avail themselves of opportunities available to them through government and other groups, and many other such social emphases, became part of our program.
The Bible is very clear that we must protect and support those who are oppressed, weak and vulnerable. I have learned that I must give this work priority in my life so that even if I am busy, I should find the time for it, even by cancelling other appointments. So during the war, when innocent YFC workers or volunteers were arrested on suspicion of being terrorists, I took it as my call to do all I can to seek their release. I sometimes had to spend as much as six hours in the police station. I would go with my books and use the time to study as I waited. We were able to secure the release of several people. There are others also who are vulnerable who needed our support—such as women who had been abused or raped, widows who have tenants who are exploiting them and people who have done wrong and are charged under the law. As they are so weak, their grievances could be dismissed through the persuasion of powerful people. So they need people to stand by them in their time of need. I have had to go to the courts on occasion because of this. I found that this could be a costly business as some people may get upset with us for standing with these vulnerable people.
Nurturing new believers is an important part of the program of an evangelistic ministry. As the biblical lifestyle includes commitment to social responsibility, this became a part of our nurturing curriculum. Some who had been nurtured in our work opted for careers in social action related causes. It has been our joy to see many youth emerging from our ministry becoming active in macro-social projects which evangelistic organizations cannot be engaged in. It has also been my joy to act as an encourager to a few biblical Christians in Sri Lanka and elsewhere who have sensed a primary call to the fields of social and political action. As for me, my primary work remains in the fields of evangelism and nurturing those resulting from and involved in it.
Yet, in the evangelical community there still remains a suspicion that comes from the old "evangelism-versus-social-concern" wars. Those of us who are involved in evangelism are still accused of being insensitive to human need and biblical Christians involved in social concern are accused of being unbiblical. For many years I have been wishing for the emergence of a crop of younger Christians belonging to the "post-war" generation that focus on obeying what the Bible says and the call God gives them without letting the prejudices of a past era cloud their thinking. I hope that Youth for Christ will help nurture many such younger Christians.
Will this emphasis on social concern result in the neglect of evangelism by the church? After all there are so many social needs in the world that even if the whole church gives itself devotedly to this work there would still be so much more to do. Where's the time or energy for evangelism? If we realize that people are lost without Christ, that would not happen. We would do all we can, amidst our busy schedules of meeting human need, to ensure that all people everywhere have an opportunity to meet their greatest human need, their greatest human right: their need and right to know the salvation their Creator has won for them.
[1] I still remain very active as a lay leader and local preacher in the church of my youth, the Methodist Church of Sri Lanka. The commitment to social action and justice in our denomination remains strong, but I rejoice in the revival of commitment to evangelism and in the growth of the church through the starting of faith communities among people previously unreached by the gospel message.