††institutetext:  aInstitute for Theoretical Physics, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany
bInstitute for Astroparticle Physics, Karlsruhe Institute of Technology,
76344 Eggenstein-Leopoldshafen, Germany
cPhysik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich,
Switzerland
dInstitute for Particle Physics Phenomenology, Department of Physics,
University of Durham, Durham, DH1 3LE, UK
eLudwig-Maximilians-Universität München, Fakultät für Physik,
Arnold Sommerfeld Center for Theoretical Physics, D-80333 München, Germany,
fTheoretical Physics Department, CERN, 1211 Geneva 23, Switzerland

# Single Photon Production at Hadron Colliders at NNLO QCD with Realistic
Photon Isolation

X. Chena,b, T. Gehrmannc, E.W.N. Gloverd, M. Höfere, A. Hussf, R. Schürmannc
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Isolated photons at hadron colliders are defined by permitting only a limited
amount of hadronic energy inside a fixed-size cone around the candidate photon
direction. This isolation criterion admits contributions from collinear photon
radiation off QCD partons and from parton-to-photon fragmentation processes.
We compute the NNLO QCD corrections to isolated photon and photon-plus-jet
production, including these two contributions. Our newly derived results allow
us to reproduce the isolation prescription used in the experimental
measurements, performing detailed comparisons with data from the LHC
experiments. We quantify the impact of different photon isolation
prescriptions, including no isolation at all, on photon-plus-jet cross
sections and discuss possible measurements of the photon fragmentation
functions at hadron colliders.

###### Keywords:

Hadron Colliders, QCD Phenomenology, Photons, Jets, NNLO Corrections

††preprint: ZU-TH 16/22
KA-TP-12-2022
P3H-22-044
IPPP/22/30
CERN-TH-2022-072
LMU-ASC 17/22

##  1 Introduction

High-transverse momentum photons are produced copiously at hadron colliders
and provide a distinctive final-state signature in the form of an
electromagnetic energy deposit. Their production cross sections have been
studied at early hadron colliders Athens-Athens-Brookhaven-CERN:1979oxx ;
Anassontzis:1982gm ; CMOR:1989qzc ; UA1:1988zam ; UA2:1991wce  as well as in
fixed-target experiments WA70:1987vvj ; FermilabE706:2004emk , followed by
first precision measurements at the Fermilab Tevatron D0:2005ofv ; D0:2008chx
; D0:2013lra ; CDF:2017cuc . Production cross sections for photons at high
transverse momentum have been measured at the LHC by the ATLAS ATLAS:2017nah ;
ATLAS:2019buk ; ATLAS:2017xqp , CMS CMS:2018qao ; CMS:2019jlq  and ALICE
ALICE:2019rtd  collaborations, either fully inclusively or in association with
hadronic jets. Besides their importance for precision QCD phenomenology, these
cross sections are also key ingredients in data-driven background estimations
in new physics searches Lindert:2017olm .

With the Born-level production process Halzen:1978rx ; Ruckl:1978mx  being
photon radiation off a quark in quark–antiquark annihilation or quark–gluon
scattering, these measurements offer direct sensitivity to the gluon
distribution in the proton. In global determinations of parton distribution
functions, the inclusion of photon production data has long been a matter of
debate Harriman:1990hi ; Vogelsang:1995bg ; dEnterria:2012kvo ;
Carminati:2012mm ; Campbell:2018wfu , relating especially to the
quantification of uncertainties associated with the identification of photons
in the experimental measurements and the implementation of photon
identification in the corresponding theory calculations. Photons that are
observed in a hadronic environment can arise from different sources: in
addition to the direct Born-level process that is calculable at parton level,
they can also be emitted in the course of the parton-to-hadron transition
(fragmentation process), described by photon fragmentation functions
Koller:1978kq ; Laermann:1982jr , or be produced as secondary photons in the
decay of hadrons. In order to single out the direct production process over
fragmentation and photons from hadron decays, observed photons are required to
be sufficiently isolated from other hadronic activity in the detector. This
isolation criterion is typically formulated in the experimental studies by
admitting only a limited amount of hadronic energy in a fixed-size cone around
the direction of the photon candidate. For a fixed-size cone, a non-zero
amount of hadronic energy must be admitted to ensure the infrared safety of
the resulting observable in QCD perturbation theory, where the phase space for
soft radiation is not allowed to be geometrically restricted. This cone-based
isolation is very efficient in suppressing secondary photons, but admits
contributions from photon fragmentation processes. Consequently, the theory
description of photon production observables must account for both the direct
and fragmentation contributions, and subject both contributions to the photon
isolation procedure applied in the experimental measurement. The fragmentation
contribution can be eliminated by imposing a dynamical isolation cone
Frixione:1998jh , smoothly lowering the hadronic energy cut towards its center
where all emissions are vetoed. While theoretically very appealing and
practical, in practice, this dynamical cone isolation could not be faithfully
implemented in experimental measurements.

With a fixed-cone isolation prescription, direct and fragmentation
contributions can not be separated from their final-state signatures and are
closely linked to each other in the theory description. In particular, higher
order perturbative terms in the direct contribution contain collinear
parton–photon configurations that give rise to infrared singularities that are
compensated by the mass factorisation terms Koller:1978kq ; Laermann:1982jr
of the parton-to-photon fragmentation functions. The next-to-leading order
(NLO) predictions for inclusive photon Aurenche:1987fs ; Baer:1990ra ;
Aurenche:1992yc ; Gordon:1993qc ; Gluck:1994iz ; Catani:2002ny ;
Catani:2013oma , and photon-plus-jet Aurenche:2006vj  production, which share
the same parton-level processes, account for these fragmentation
contributions. They depend on the photon fragmentation functions, which fulfil
inhomogeneous DGLAP-type evolution equations with a priori unknown non-
perturbative boundary conditions. The available parameterisations of the
photon fragmentation functions Owens:1986mp ; Gluck:1992zx ; Bourhis:1997yu
rely on model estimates for these boundary conditions. Measurements of non-
isolated photon production at LEP Buskulic:1995au ; Ackerstaff:1997nha  are
the only data that enable a determination of the photon fragmentation
functions GehrmannDeRidder:1997gf  and a critical assessment
GehrmannDeRidder:1998ba  of the model estimates.

Up to now, next-to-next-to-leading order (NNLO) calculations Campbell_2017 ;
Campbell:2017dqk ; Chen_2020  of isolated photon and photon-plus-jet
production were only performed for a dynamical cone isolation (or a hybrid
variant thereof Siegert:2016bre ), since none of the available infrared
subtraction schemes was able to handle fragmentation processes at NNLO. This
conceptual limitation has been overcome recently with the incorporation of
heavy hadron fragmentation processes into residue subtraction Czakon:2021ohs
and photon fragmentation processes in antenna subtraction Gehrmann:2022cih .
In this work, we employ the antenna subtraction method GehrmannDeRidder:2005cm
; Daleo:2006xa ; Currie:2013vh  to compute the NNLO corrections to isolated
photon and photon-plus-jet observables.

The paper is structured as follows. Section 2 reviews the different photon
isolation prescriptions used in experiment and theory and discusses their
effect on direct and fragmentation contributions. The calculation of the NNLO
corrections and their implementation in the NNLO  JET parton-level event
generator is discussed in Section 3. Our newly derived predictions are
compared with data from the LHC experiments in Section 4, discussing in
particular the impact on precision phenomenology applications. The
incorporation of fragmentation processes allows us to perform a detailed
comparison of different isolation prescriptions in Section 5, where we also
quantify the numerical impact of the photon fragmentation functions. In view
of a potential hadron collider measurement of the photon fragmentation
functions, we introduce a new observable derived from photon-plus-jet final
states in Section 6, and discuss its behaviour for different photon isolation
prescriptions. Section 7 summarises our most important findings and provides
an outlook on future precision phenomenology with isolated photon observables.

##  2 Definition of Isolated Photons

Photons can be produced by three different mechanisms in hadronic collisions.
Direct photons originate from the point-like QED coupling to quarks and are
produced in the hard scattering event. These photons usually have a clean
signature, being well separated from additional hadronic radiation. The second
class of photons are photons from fragmentation, i.e. these photons are
radiated during the hadronisation process of an ordinary jet production event.
This fragmentation process is described by non-perturbative parton-to-photon
fragmentation functions $D_{p\to\gamma}$. Both direct and fragmentation
photons are primary photons. The last class of photons are secondary photons,
which result from hadronic decays (e.g. $\pi^{0}\to\gamma\gamma$). The
majority of photons produced in hadronic collisions are secondary photons. In
order to suppress this overwhelming background of photons and enhance the
contribution from direct photons, a photon isolation criterion is imposed in
the experimental measurement. Isolation criteria limit the hadronic energy in
the vicinity of the photon by a maximal value $E_{T}^{\rm max}$. It is noted
that lowering the maximal hadronic energy to zero is not feasible from an
experimental point of view, nor it is infrared safe since soft gluon emission
must be unrestricted across the entire phase space to cancel infrared
singularities. Different isolation prescriptions that have been used in
experimental analyses and theoretical calculations are summarised as the
following:

  * 1.

Fixed Cone Isolation: The hadronic energy inside a cone around the photon
direction with radius $R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ in
pseudorapidity $\eta$ and azimuthal angle $\phi$ is integrated. If the total
hadronic energy $E_{T}^{\rm hadr}$ does not exceed a certain maximal value
$E_{T}^{\rm max}$ the photon is considered to be isolated. The maximal
hadronic energy can depend on the photon transverse momentum and is commonly
parametrised in the form

$E_{T}^{\rm max}=\varepsilon\,p_{T}^{\gamma}+E_{T}^{\rm thres}\,.$ (1)

The fixed cone isolation is characterised by three parameters
$(R,\,\varepsilon,\,E_{T}^{\rm thres})$ and is used in all experimental
measurements of photon production cross sections to date.

  * 2.

Smooth Cone Isolation Frixione:1998jh : All cones with radii $r_{d}$ smaller
than a maximal cone radius $R_{d}$ are considered. A photon candidate
satisfies the smooth cone isolation criterion if

$E_{T}^{\rm hadr}(r_{d})<E_{T}^{\rm max}(r_{d})\quad\forall\,r_{d}\leq
R_{d}\,.$ (2)

The maximal hadronic energy is smoothly decreasing with smaller cone radii
$r_{d}$ and vanishes for $r_{d}=0$. This behaviour can be implemented using
the functional form

$E_{T}^{\rm max}(r_{d})=\varepsilon_{d}\,p_{T}^{\gamma}\left(\frac{1-\cos
r_{d}}{1-\cos R_{d}}\right)^{n}\,,$ (3)

where $\varepsilon_{d}>0$ and $n>0$ are free parameters. As exact collinear
radiation in the photon direction is vetoed by the smooth cone isolation
criterion, the fragmentation contribution is completely eliminated. The smooth
cone isolation is characterised by three parameters
$(R_{d},\,\varepsilon_{d},\,n)$. Due to the finite resolution from the cell
size of the electromagnetic calorimeter, the smooth cone isolation can only be
approximated in an experimental analysis and so far has not been used in any
measurement.

  * 3.

Hybrid Isolation Siegert:2016bre : A smooth cone isolation and a fixed cone
isolation are combined. First a smooth cone isolation with radius $R_{d}$ is
applied to the event. By doing so, radiation collinear to the photon is
completely vetoed and the fragmentation contribution is eliminated. To all
events which pass the smooth cone isolation a fixed cone isolation with a cone
radius $R$ is applied. The two cone radii are chosen in such a way that
$R>R_{d}$. Outside of the inner cone, hadronic radiation is treated in the
same manner as in the fixed cone isolation. Therefore, the hybrid isolation is
expected to be a better approximation of the fixed cone isolation compared to
the smooth cone isolation.

  * 4.

Democratic Clustering Glover:1993xc ; GehrmannDeRidder:1997gf : The photon
candidate is clustered together with the partons by using a sequential jet
clustering algorithm. Upon completion of the clustering, the cluster
containing the photon candidate is identified and the fraction of
electromagnetic energy inside the cluster

$z_{\rm em}=\frac{p_{T}^{\gamma}}{E_{T}^{\rm jet}}$ (4)

is determined. If this energy fraction is larger than a certain minimal value
$z_{\rm cut}$ the photon is considered to be isolated. The democratic
clustering isolation is described by the jet algorithm and the parameter
$z_{\rm cut}$. Refinements to the democratic clustering approach using modern
jet substructure techniques have been discussed recently Hall:2018jub .

Having implemented fragmentation processes into the theory predictions, it is
also possible to obtain results for the photon production cross section
without imposing any photon isolation. These inclusive predictions contain
photons from primary production processes. Secondary photons, which would
dominate the measured non-isolated photon production cross section, are not
included. Therefore, these inclusive predictions can not directly be compared
to experimental measurements. It is however noted, that they can still provide
important insights on direct and fragmentation photons in inclusive scattering
kinematic. A detailed numerical study of the impact of the isolation
prescription on the photon-plus-jet cross section is presented in Section 5,
where also results for non-isolated photon-plus-jet production are shown.

##  3 Computational Set-Up

Isolated photon and photon-plus-jet production at hadron colliders originate
from the same parton-level subprocesses, since the finite transverse momentum
of the isolated photon requires the presence of a partonic recoil. Isolated
photon cross sections are fully inclusive over this extra partonic radiation,
while photon-plus-jet cross sections are obtained by requiring this extra
radiation to be recombined into a resolved jet within kinematical acceptance.
The relevant Born-level subprocesses Halzen:1978rx ; Ruckl:1978mx  are
quark–antiquark annihilation $q\bar{q}\to\gamma g$ and quark–gluon scattering
$qg\to\gamma q$.

Three different parton-level contributions combine to yield the NNLO QCD
corrections for isolated photon and photon-plus-jet production: the double
virtual contribution (VV) consisting of the two-loop corrections to the Born-
level processes, the real-virtual contribution (RV) consisting of one-loop
Feynman diagrams with an additional parton radiation and double real (RR)
contribution where two additional partons are emitted. All matrix elements in
the different types of contributions are known in compact analytic form
Anastasiou:2001sv ; Bern:2003ck ; Bern:1994fz ; Signer:1995np ; Signer:1995a ;
DelDuca:1999pa .

All three types of NNLO corrections are individually infrared divergent. The
RR and RV contributions exhibit singularities from configurations in which
additional partons become unresolved, i.e. soft and/or collinear. The RV and
VV corrections contain explicit singularities from the integration over the
corresponding loop momenta. These singularities cancel among each other once
all contributions are summed up and the mass factorisation related to the
parton distributions and fragmentation functions is performed. In order to
extract the singularities in the different types of corrections and combine
them into an explicitly finite result, a subtraction method is employed. For
our calculation we use the antenna subtraction method GehrmannDeRidder:2005cm
; Daleo:2006xa ; Currie:2013vh , which is implemented in the NNLO  JET parton-
level event generator.

We present in this paper the first NNLO calculation using the fixed cone
photon isolation that is applied in all experimental analyses to date.
Therefore, it overcomes an essential drawback of previous NNLO calculations
for inclusive photon Campbell_2017 ; Chen_2020  and photon-plus-jet production
Chen_2020 ; Campbell:2017dqk  that used idealised isolations (smooth cone or
hybrid isolation), which rely on an empirical tuning of the isolation
parameters in the theory calculation to approximate the effect of the
experimental fixed cone procedure. A fixed cone isolation allows singular
photon-parton collinear configurations, which have to be subtracted along with
the QCD singularities up to NNLO. At variance with the subtraction of QCD
singularities for which no information on individual partons in unresolved
limits is required, photonic singularities have to be subtracted while
retaining the information on the energy fraction of the photon in the
collinear cluster. For the subtraction of photonic singularities we use a new
class of antenna functions, fragmentation antenna functions Gehrmann:2022cih ,
whose integrated forms depend on the final-state momentum fraction.
Calculations with a fixed cone isolation also receive contributions from
fragmentation processes. The mass factorisation terms of the fragmentation
functions cancel the explicit poles of the integrated fragmentation antenna
functions. Our construction of the antenna subtraction terms relevant to
photon-parton collinear singularities and their interplay with the mass
factorisation of the fragmentation functions is described in detail elsewhere
Gehrmann:2022cih .

The additional subtraction terms needed for the subtraction of the
parton–photon collinear limits are largely independent of the genuine QCD
subtraction terms, so that they can be combined in an additive manner. For the
calculation presented in this paper we build on the QCD subtraction terms
which were already used in a previous NNLO  JET calculation for photon
production cross sections Chen_2020  with a hybrid isolation.

The fragmentation contribution as well as the direct contribution (through the
mass factorisation terms of the fragmentation functions) depend on the
fragmentation scale $\mu_{A}$. In the same manner as for the mass
factorisation scale in the parton distribution functions, this scale should be
chosen close to the characteristic scale of the process under consideration.
In the presence of photon isolation, the determination of this characteristic
scale does however deserve special attention. The mass factorisation procedure
dresses the bare parton distributions and fragmentation functions with
unresolved parton radiation. This unresolved parton radiation is inclusive in
transverse momentum relative to the primary bare parton, with relative
transverse momenta ranging between zero (the exact collinear limit) and the
factorisation scale. This factorisation scale does therefore reflect up to
which relative transverse momentum scale additional parton radiation is
admitted into the observable under consideration. For the mass factorisation
of the parton distributions, this factorisation scale can clearly be chosen to
be of the order of the transverse momentum of the photon $p_{T}^{\gamma}$,
since the isolated photon and photon-plus-jet cross sections are inclusive in
initial-state radiation up to this scale. The situation for final-state photon
fragmentation is however entirely different. Here, the relative transverse
momentum of the final-state photon with respect to the fragmenting
parton–photon cluster defines the characteristic scale of the process. In the
absence of any isolation criterion, this scale could also be of the order of
$p_{T}^{\gamma}$, since no restrictions on additional radiation are applied.
As soon as an isolation requirement is imposed on the photon, however, any
additional radiation associated with the fragmentation process is severely
restricted. Only radiation inside the isolation cone and below the maximal
hadronic energy threshold is admitted, thereby substantially lowering the
scale up to which the observable is inclusive in extra radiation.

For our calculations we set this scale close to the maximal invariant mass
inside the isolation cone. For a fixed cone isolation with cone radius $R$ and
maximal hadronic energy $E_{T}^{\rm max}$ it reads

$m_{\rm cone}=\sqrt{E_{T}^{\rm
max}\,p_{T}^{\gamma}\,R^{2}}+\mathcal{O}(R^{3})$ (5)

where we assumed that the hadronic energy stems from a single parton radiation
at the distance $R$ to the photon. This assumption is valid as long as the
value for the scale remains larger than the hadronisation scale. Due to the
small values for $E_{T}^{\rm max}$ used in the experimental isolation
prescriptions the typical numerical values for the fragmentation scale are
$\mathcal{O}(1\,\mathrm{GeV})$ to $\mathcal{O}(10\,\mathrm{GeV})$ and
therefore small compared to the renormalisation and factorisation scale which
are set by default to $\mu_{R}=\mu_{F}=p_{T}^{\gamma}$ in our calculation.

For an isolated photon cross section, a choice of fragmentation scale
$\mu_{A}$ of the order of $p_{T}^{\gamma}$ would imply that partonic radiation
up to this scale is removed from the direct contribution and assigned to the
fragmentation contribution, which is added back in a resummed form. For
photon-parton invariant masses between $m_{\rm cone}$ and $p_{T}^{\gamma}$,
this radiation is however either vetoed by the isolation prescription or
represents resolved out-of-cone configurations that should remain part of the
direct contribution. By attributing them to the fragmentation contribution,
their kinematics is misrepresented and their overall impact is underestimated
since the collinear resummation of the photon fragmentation functions
attenuates them in the range of momentum fractions relevant to isolated photon
cross sections. As a matter of fact, cone-isolation cross sections lead to a
complicated hierarchy of scales Catani:2013oma  and to the emergence of non-
global logarithmic corrections Balsiger:2018ezi ; Ebert:2019zkb  that
potentially require dedicated resummation.

The scale uncertainties of the predictions are determined by varying the three
scales by a factor of 2 and $\tfrac{1}{2}$ around their central values:

$\mu_{R}=a\,p_{T}^{\gamma}\,,\quad\mu_{F}=b\,p_{T}^{\gamma}\,,\quad\mu_{A}=c\,m_{\rm
cone}\,,$ (6)

with $a,b,c\in\bigl{\\{}\tfrac{1}{2},1,2\bigr{\\}}$ and only retaining
combinations with $\tfrac{1}{2}\leq a/b,\,a/c,\,b/c\leq 2$ (15-point scale
variation). For our predictions we use the NNPDF4.0 PDF set NNPDF:2021njg  and
the BFGII set Bourhis:1997yu  for the parton-to-photon fragmentation
functions. Moreover, we assign an $\mathcal{O}(\alpha)$ power counting to the
quark-to-photon and gluon-to-photon fragmentation functions, which is
justified by the low mass factorisation scale used for these functions, and
their overall small contribution to the cross sections. It should be noted
that this power counting differs from the usual assignment of
$\mathcal{O}(\alpha/\alpha_{s})$ to the photon fragmentation functions, which
lacks however any formal ground as soon as $\mu_{A}\neq\mu_{R}$.

Imposing this power counting, the photon production cross sections at the
different levels of accuracy read

$\displaystyle\mathrm{d}\hat{\sigma}^{\gamma+X,{\rm LO}}$ $\displaystyle=$
$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm LO}\,,$
$\displaystyle\mathrm{d}\hat{\sigma}^{\gamma+X,{\rm NLO}}$ $\displaystyle=$
$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm
NLO}+\mathrm{d}\hat{\sigma}^{\rm NLO}_{\rm frag}\,,$
$\displaystyle\mathrm{d}\hat{\sigma}^{\gamma+X,{\rm NNLO}}$ $\displaystyle=$
$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm
NNLO}+\mathrm{d}\hat{\sigma}_{\rm frag}^{\rm NNLO}\,,$ (7)

where the direct contribution is given by

$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm LO}$ $\displaystyle=$
$\displaystyle\,\mathrm{d}\hat{\sigma}_{\gamma}^{{\rm LO}},$
$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm NLO}$ $\displaystyle=$
$\displaystyle\mathrm{d}\hat{\sigma}_{\gamma}^{{\rm
NLO}}+\mathrm{d}\hat{\sigma}_{\rm MF}^{{\rm NLO}}\,,$
$\displaystyle\mathrm{d}\hat{\sigma}_{\rm dir}^{\rm NNLO}$ $\displaystyle=$
$\displaystyle\mathrm{d}\hat{\sigma}_{\gamma}^{{\rm
NNLO}}+\mathrm{d}\hat{\sigma}_{\rm MF}^{\rm NNLO}\,,$ (8)

and the fragmentation contribution at the respective order takes the form

$\displaystyle\mathrm{d}\hat{\sigma}_{\rm frag}^{\rm NLO}$ $\displaystyle=$
$\displaystyle\sum_{p}\mathrm{d}\hat{\sigma}^{{\rm LO}}_{p}\otimes
D_{p\to\gamma}\,,$ $\displaystyle\mathrm{d}\hat{\sigma}_{\rm frag}^{\rm NNLO}$
$\displaystyle=$ $\displaystyle\sum_{p}\mathrm{d}\hat{\sigma}_{p}^{{\rm
NLO}}\otimes D_{p\to\gamma}\,.$ (9)

Here, the sum over $p$ runs over all final-state partons and $D_{p\to\gamma}$
are the mass-factorised fragmentation functions. The cross section
$\mathrm{d}\hat{\sigma}_{i}$ denotes the production of a hard resolved
particle $i$ and $\mathrm{d}\hat{\sigma}_{\rm MF}$ contains the mass
factorisation counter-terms of the fragmentation functions.

###  3.1 Comparison with JETPHOX

NLO QCD corrections to isolated photon and photon-plus-jet cross sections
using a fixed cone isolation have already been calculated and implemented in
the JETPHOX code Catani:2002ny . These predictions allow for a validation of
the implementation of fragmentation processes up to NNLO (employing our
counting of orders) and the validation of the direct contribution up to NLO in
NNLO  JET .

In order to compare with JETPHOX we use a similar set-up to the ATLAS 13
$\mathrm{TeV}$ photon-plus-jet study ATLAS:2017xqp . The following cuts on the
transverse momenta and (pseudo-)rapidities of the photon and the jet are
imposed:

$p_{T}^{\gamma}>125\,\mathrm{GeV}\quad p_{T}^{{\rm
jet}}>100\,\mathrm{GeV},\quad|\eta_{\gamma}|<2.37,\quad|y_{{\rm
jet}}|<2.37\,.$ (10)

To simplify the set-up we do not apply additional cuts on the pseudorapidity
of the photon nor do we require a minimum angular separation between the
photon and the jet. A fixed cone photon isolation with the parameters

$R=0.4,\quad\varepsilon=0.0042,\quad E_{T}^{\rm thres}=10\,\mathrm{GeV}$ (11)

is imposed. Jets are constructed using a $k_{T}$ jet algorithm with a cone
radius $R_{\rm jet}=0.4$. For comparison only we set all scales (also the
fragmentation scale) to the transverse momentum of the photon, i.e.
$\mu_{R}=\mu_{F}=\mu_{A}=p_{T}^{\gamma}$ and use the NNPDF31_nnlo_as_0118_mc
PDF set NNPDF:2017mvq .

| NNLO  JET | JETPHOX
---|---|---
$\sigma_{{\rm dir}}^{{\rm LO}}$ | $(192.487\pm 0.009)\,\mathrm{pb}$ | $(192.488\pm 0.003)\,\mathrm{pb}$
$\sigma_{{\rm dir}}^{{\rm NLO,only}}$ | $(97.903\pm 0.058)\,\mathrm{pb}$ | $(97.75\pm 0.14)\,\mathrm{pb}$
$\sigma_{{\rm frag}}^{{\rm NLO}}$ | $(14.255\pm 0.009)\,\mathrm{pb}$ | $(14.2550\pm 0.0002)\,\mathrm{pb}$
$\sigma_{{\rm frag}}^{{\rm NNLO,only}}$ | $(14.999\pm 0.022)\,\mathrm{pb}$ | $(14.971\pm 0.015)\,\mathrm{pb}$
Table 1: Comparison of direct and fragmentation contributions to the photon-
plus-jet cross section obtained with NNLO  JET and JETPHOX. The ’only’ numbers
indicate the increment from the coefficient at the respective order.

The comparison between the JETPHOX and NNLO  JET results for the predictions
of the different contributions to the photon-plus-jet cross sections are shown
in Table 1. The predictions for the direct and fragmentation contribution
obtained with NNLO  JET and JETPHOX agree within the respective integration
uncertainties. However, we discovered a 1% deviation between the NNLO
fragmentation coefficient stemming from the numerically small gluon-to-photon
fragmentation process alone, which is however within the technical
uncertainties of the JETPHOX implementation.

##  4 Comparison with Experimental Measurements

The cross section for photon production in hadronic collisions has been
studied extensively in experimental measurements. Earlier measurements of
isolated photon Athens-Athens-Brookhaven-CERN:1979oxx ; Anassontzis:1982gm ;
CMOR:1989qzc ; UA1:1988zam ; UA2:1991wce ; D0:2005ofv ; CDF:2017cuc  and
photon-plus-jet production D0:2008chx ; D0:2013lra  in hadronic collisions
were followed by measurements at the LHC. These measurements for isolated
photon production ATLAS:2017nah ; CMS:2018qao ; ATLAS:2019buk ; ALICE:2019rtd
as well as for the production of a photon in association with a jet
CMS:2018qao ; ATLAS:2017xqp ; CMS:2019jlq  are covering a large kinematic
range and are reaching percent level accuracy demanding for theory predictions
meeting this level of precision.

In the experimental analyses selection cuts on the photon are imposed.
Typically these cuts limit the transverse momentum $p_{T}^{\gamma}$ and the
pseudorapidity $\eta_{\gamma}$ of the photon to a specific kinematic range.
Photon-plus-jet measurements use the anti-$k_{T}$ algorithm Cacciari:2008gp
for jet reconstruction and apply additional cuts on the transverse momentum of
the (leading) jet $p_{T}^{\rm jet}$ as well as on its rapidity $y_{\rm jet}$
or pseudorapidity $\eta_{\rm jet}$.

In the following we will compare our NNLO predictions for isolated photon
production to 13 $\mathrm{TeV}$ ATLAS ATLAS:2019buk  and 7 $\mathrm{TeV}$
ALICE data ALICE:2019rtd  and our predictions for the photon-plus-jet cross
section to 8 $\mathrm{TeV}$ CMS CMS:2019jlq  and 13 $\mathrm{TeV}$ ATLAS data
ATLAS:2017xqp . Throughout this comparison we set the value for the
electromagnetic coupling constant to $\alpha=\alpha(0)=1/137.036$, as is
appropriate for the coupling of an external resolved photon. For the first
time these NNLO predictions employ the same photon isolation as in the
experimental measurements, thus avoiding the mismatch inherent in previous
theory–data comparisons at this order and eliminating the associated source of
systematic uncertainty.

###  4.1 ATLAS 13 TeV Isolated Photon Measurement

The ATLAS 13 $\mathrm{TeV}$ isolated photon measurement ATLAS:2019buk
provides data for the photon production cross section differential in the
photon transverse momentum $p_{T}^{\gamma}$ for four different pseudorapidity
bins

$|\eta_{\gamma}|<0.6,\quad 0.6<|\eta_{\gamma}|<1.37,\quad
1.56<|\eta_{\gamma}|<1.81,\quad 1.81<|\eta_{\gamma}|<2.37\,.$ (12)

The photon is required to have a minimum transverse momentum
$p_{T}^{\gamma}>125\,\mathrm{GeV}$. Moreover, a fixed cone photon isolation
with parameters

$R=0.4,\quad\varepsilon=0.0042,\quad E_{T}^{\rm thres}=4.8\,\mathrm{GeV}$ (13)

is imposed. For our theoretical predictions we use the same photon isolation
and the scales are set according to (6).

Figure 1: Photon transverse momentum distributions for isolated photon
production at LO, NLO and NNLO for four different pseudorapidity bins. The
prediction are compared to 13 $\mathrm{TeV}$ ATLAS data ATLAS:2019buk .

Figure 1 shows the isolated photon cross section for the four photon
pseudorapidity bins. For all bins the NNLO corrections are positive and
increase the cross section by 5–6% compared to the NLO cross section. These
corrections are relatively constant over the whole transverse momentum range.
However, we observe an increase in the NNLO-to-NLO K-factor for the first two
pseudorapidity bins for very large transverse momenta where the corrections
increase to roughly 9% compared to the NLO predictions. In contrast, for the
forward pseudorapidity bins the K-factor is decreasing for large transverse
momenta where the difference between the NLO and NNLO cross section is only a
few percent.

The scale uncertainty at NLO is 18% for the first and second pseudorapidity
bin and increases to 24% for the more forward pseudorapidity bins. In all bins
the scale uncertainty is slightly increasing towards larger $p_{T}^{\gamma}$.
Including NNLO corrections significantly reduces the scale uncertainty. Over
most of the range in $p_{T}^{\gamma}$ and $\eta_{\gamma}$ the NNLO scale
variation amounts to $(+1.5,-3.0)\%$ and it only increases moderately for
large photon transverse momenta.

Comparing our NNLO predictions with the ATLAS data, we observe that in the
first, third and fourth pseudorapidity bin the slope of the measured
distribution is well captured by the predictions. However, the predictions
undershoot the data by 10% (15%) in the first (third) bin and are not
compatible in normalisation with the data over the largest range of photon
transverse momenta. In the second and fourth pseudorapidity bin, data and
predictions agree in the mid $p_{T}^{\gamma}$ region. In the second bin, the
slope of the measured distribution is slightly steeper than that of the theory
predictions.

The fragmentation contribution to the NNLO cross section is very small and
amounts to only 2% of the full NNLO cross section in the low $p_{T}^{\gamma}$
region and is negligible for mid and large transverse momenta. Therefore, the
NNLO predictions presented here are very similar to the NNLO predictions
obtained with a hybrid isolation shown in ATLAS:2019buk , taking however into
consideration a different overall normalisation due to the different value for
the electromagnetic coupling constant that was used there.

###  4.2 ALICE 7 TeV Isolated Photon Measurement

The ALICE 7 $\mathrm{TeV}$ isolated photon measurement ALICE:2019rtd  provides
data for the photon production cross section differential in the photon
transverse momentum in the range
$10\,\mathrm{GeV}<p_{T}^{\gamma}<60\,\mathrm{GeV}$. The photon is required to
have a pseudorapidity $|\eta_{\gamma}|<0.27$ and a fixed cone photon isolation
with the parameters

$R=0.4,\quad\varepsilon=0,\quad E_{T}^{{\rm thres}}=2\,\mathrm{GeV}$ (14)

is imposed.

Figure 2: Predictions for the photon transverse momentum distribution at LO,
NLO, and NNLO. The predictions are compared to data from the ALICE measurement
ALICE:2019rtd .

The prediction for the photon transverse momentum distribution and the
comparison to the ALICE data is shown in Figure 2. In our predictions for the
ALICE measurement we used a fixed fragmentation scale of
$\mu_{A}=3\,\mathrm{GeV}$ since our default choice $\mu_{A}=m_{\rm cone}$ in
Eq. (5) yields numerical values for the fragmentation scale that are too small
and outside of the supported range of the BFGII fragmentation functions.
Lastly, we multiply our predictions by the correction factors accounting for
hadronisation effects and underlying event contributions that are quoted in
ALICE:2019rtd . We use a fixed cone photon isolation with the parameters of
(14).

The NNLO corrections enhance the NLO predictions by approximately 8% in the
region $10\,\mathrm{GeV}<p_{T}^{\gamma}<16\,\mathrm{GeV}$ and 6% in the region
$16\,\mathrm{GeV}<p_{T}^{\gamma}<60\,\mathrm{GeV}$. A significant reduction of
scale uncertainty is observed when going from NLO to NNLO. The NLO scale
variation ranges from 54% in the lowest bin to 26% in the highest bin. In
contrast, the scale uncertainty at NNLO is only $(+5,-10)\%$ in the low
$p_{T}^{\gamma}$ regime and further decreases to $(+1.5,-4.5)\%$ in the last
transverse momentum bin.

All data points are in agreement with our NNLO prediction within the
experimental and theoretical uncertainties, noting that experimental errors
exceed the scale uncertainty of the NNLO calculation across the entire
kinematic range.

The contribution from fragmentation processes to the NNLO cross section is
decreasing with increasing $p_{T}^{\gamma}$. In the lowest bin the
fragmentation contribution amounts to 10% of the cross section while in the
highest bin it only contributes 2% to the cross section.

###  4.3 CMS 8 TeV Photon-Plus-Jet Measurement

The CMS 8 $\mathrm{TeV}$ photon-plus-jet study CMS:2019jlq  provides data for
the photon-plus-jet cross section as a function of the transverse momentum of
the photon, the pseudoradidity of the photon, and the pseudorapidity of the
jet. The photon is required to have $p_{T}^{\gamma}>40\,\mathrm{GeV}$ and
$|\eta_{\gamma}|<1.44$ or $1.57<|\eta_{\gamma}|<2.5$. Jets are identified
using the anti-$k_{T}$ algorithm with a cone radius $R_{{\rm jet}}=0.5$. The
leading jet is required to have $p_{T}^{{\rm jet}}>25\,\mathrm{GeV}$ and
$|\eta_{{\rm jet}}|<2.5$. Moreover, it must be separated from the photon
candidate by $\Delta R_{\gamma{\rm jet}}>0.5$.

While the ATLAS and ALICE measurements apply their photon isolation procedures
at the fiducial cross section level, photon isolation in the CMS measurement
CMS:2019jlq  is performed only at the level of object reconstruction. The
photon isolation parameters used in the CMS measurement can thus not be taken
over to the theory predictions (which are always at the level of fiducial
cross sections) in a direct manner. The CMS 8 $\mathrm{TeV}$ study recommends
CMS:2019jlq  that theory predictions should be computed with the fixed-cone
isolation parameters

$R=0.2,\quad\varepsilon=0,\quad E_{T}^{{\rm thres}}=4\,\mathrm{GeV}$ (15)

to mimic the effect of photon isolation in their measurement.

Data for the photon transverse momentum distribution are given for different
bins in the pseudorapidity of the photon

$|\eta_{\gamma}|<0.8,\quad 0.8<|\eta_{\gamma}|<1.44,\quad
1.57<|\eta_{\gamma}|<2.1,\quad 2.1<|\eta_{\gamma}|<2.5\,,$ (16)

and the jet

$|\eta_{{\rm jet}}|<0.8,\quad 0.8<|\eta_{{\rm jet}}|<1.5,\quad 1.5<|\eta_{{\rm
jet}}|<2.1,\quad 2.1<|\eta_{{\rm jet}}|<2.5\,.$ (17)

Figure 3: Photon transverse momentum distributions for different bins in the
pseudorapidity of the photon and the jet. The predictions are compared to CMS
8 $\mathrm{TeV}$ data CMS:2019jlq .

For the theory predictions we use a fixed cone isolation with the parameters
of (15). Since the numerical values of our default choice of the fragmentation
scale $\mu_{A}=m_{\rm cone}$ are again too small and not supported by the
BFGII set for these isolation parameters, we use a fixed fragmentation scale
of $\mu_{A}=3\,\mathrm{GeV}$. Our NNLO predictions are compared to CMS data in
the 16 different pseudorapidity ranges in Figure 3. For most of the range of
photon and jet pseudorapidities the NNLO corrections are positive and increase
the cross section by $7-10\%$. Compared to NLO the shape of the NNLO
distribution is slightly modified. In the low $p_{T}^{\gamma}$ regime the
corrections are small (for some pseudorapidity bins even negative) and
increase towards larger transverse momenta. For some of the forward
pseudorapidity bins a decline of the NNLO-to-NLO K-factor for high photon
transverse momenta is observed. The scale uncertainties at NLO amount to
approximately 25% for similar photon and jet pseudorapidities and increase up
to 40% for unbalanced pseudorapidity configurations. At NNLO the scale
uncertainty is reduced to $(+1,-3)\%$ for comparable pseudorapidities of the
photon and the jet. For less balanced configurations we also observe a
substantial reduction of scale uncertainty compared to NLO for small and mid
$p_{T}^{\gamma}$. However, in these configurations the NNLO scale uncertainty
increases in the high $p_{T}^{\gamma}$ regime.

Comparing the NNLO predictions to the CMS data, we observe a disagreement,
which is most prominent in the low $p_{T}^{\gamma}$ regime where the
predictions overshoot the data by up to 50%. This tension between data and
theory predictions is reduced for mid and large photon transverse momenta.
However, also in this $p_{T}^{\gamma}$ range we observe an inconsistency
between theory and data in some of the forward pseudorapidity bins of the
photon and the jet. We note that the reduction by more than a factor of two of
the scale uncertainty bands going from NLO to NNLO is essential to fully
expose this tension.

Due to the tight isolation criterion (15) the fragmentation contribution to
the NNLO cross section is small. Its contribution to the cross section is 3–5%
in the lowest photon transverse momentum bin and declines steeply for
increasing $p_{T}^{\gamma}$. However, given that these fiducial-level photon
isolation criteria are only an approximation to the object-level isolation
criteria used in the CMS measurement CMS:2019jlq , this quantification of the
fragmentation contributions is quite uncertain.

###  4.4 ATLAS 13 TeV Photon-Plus-Jet Measurement

The ATLAS 13 $\mathrm{TeV}$ photon-plus-jet measurement ATLAS:2017xqp
provides data for the cross section differential in the photon transverse
momentum, the leading jet transverse momentum, the azimuthal angular
separation between the photon and the jet, the photon–jet invariant mass, and
the cosine of the scattering angle in the photon–jet centre-of-mass system
$\cos\theta^{*}$. Jets are identified using the anti-$k_{T}$ algorithm with a
cone radius $R_{{\rm jet}}=0.4$. In the measurement the leading jet fulfilling
$|y_{{\rm jet}}|<2.37$ and $\Delta R_{\gamma{\rm jet}}>0.8$ is selected and it
has to have a transverse momentum $p_{T}^{{\rm jet}}>100\,\mathrm{GeV}$.
Photons are required to have a minimum transverse momentum of
$p_{T}^{\gamma}>125\,\mathrm{GeV}$ and they have to satisfy
$|\eta_{\gamma}|<2.37$. Moreover, photons in the transition region between the
barrel and the endcap calorimeters ($1.37<|\eta_{\gamma}|<1.56$) are excluded.
A photon isolation with parameters

$R=0.4,\quad\varepsilon=0.0042,\quad E_{T}^{\rm thres}=10\,\mathrm{GeV}$ (18)

is imposed. For the invariant mass and the $|\cos\theta^{*}|$ distribution the
additional cuts

$|\eta_{\gamma}+y_{\rm jet}|<2.37,\quad|\cos\theta^{*}|<0.83,\quad
m_{\gamma{\rm jet}}>450\,\mathrm{GeV}$ (19)

are applied. Our NNLO predictions and the comparison to the ATLAS data for the
different distributions are shown in Figure 4–8.

The $p_{T}^{\gamma}$ distribution is shown in Figure 4. Including NNLO
corrections strongly impacts the shape of the distribution compared to NLO: In
the low $p_{T}^{\gamma}$ region the cross section is reduced by $-7\%$ while
in the medium to large photon transverse momentum regime the NNLO corrections
are positive and amount to approximately $+7\%$. The NLO scale uncertainty of
$(17-22)\%$ is reduced to $(+1,-4)\%$ for low and $(+2,-5)\%$ for high photon
transverse momenta at NNLO.

Figure 4: Predictions for the photon transverse momentum distributions at LO,
NLO, and NNLO. The predictions are compared to data from the ATLAS measurement
ATLAS:2017xqp . Figure 5: Predictions for the leading jet transverse momentum
distributions at LO, NLO, and NNLO. The predictions are compared to data from
the ATLAS measurement ATLAS:2017xqp .

Comparing our NNLO predictions to data, we see excellent agreement for the
shape of the distributions. In the low and mid $p_{T}^{\gamma}$ region the
data falls slightly above the NNLO prediction. However, theory and data remain
consistent within the respective uncertainties in this regime. In the highest
bin for which data is available, the NNLO prediction overestimates the data.
This tension could indicate the onset of electroweak Sudakov logarithms
Becher:2012xr ; Becher:2013zua , which become sizeable in this region of phase
space. The contribution from fragmentation processes to the full NNLO cross
section is 5% in the low $p_{T}^{\gamma}$ regime. For larger values of the
photon transverse momentum this contribution declines strongly. A more
detailed discussion of the different contributions to the NNLO cross section
will be provided in Section 5.2 below.

The leading jet transverse momentum distribution is shown in Figure 5. At LO
the jet recoils against the photon alone and therefore the transverse momenta
of the jet and the photon coincide. Beyond LO this one-to-one correspondence
does no longer hold and the phase space region $100\,\mathrm{GeV}<p_{T}^{\rm
jet}<125\,\mathrm{GeV}$, which is not accessible at LO, is populated. This
region of phase space is contained in the first bin
($[100,130]\,\mathrm{GeV}$) of the distribution. In this bin the NNLO
corrections enhance the NLO cross section by 15% and reduce the scale
uncertainty from 28% to 12%. Since the NNLO corrections are effectively only
of NLO-type in the bulk of phase space that contributes to this bin, the scale
uncertainty remains rather large. For larger $p_{T}^{\rm jet}$ the NNLO
corrections are negative and reduce the cross section by between $-4\%$ in the
second bin and $-40\%$ in the very high $p_{T}^{\rm jet}$ region. Especially
for small and intermediate jet transverse momenta the scale uncertainty is
strongly reduced by including NNLO corrections. In the second bin the NNLO
scale uncertainty is only $(+1,-3)\%$ and it increases moderately to
$(+2,-11)\%$ in the region around $p_{T}^{\rm jet}\approx 500\,\mathrm{GeV}$.
For even larger transverse momenta the scale uncertainty increases
significantly. This kinematic regime is dominated by events with two hard
recoiling jets and a relatively soft photon. These configurations are
effectively described at NLO accuracy resulting in increasing scale
uncertainties.

Figure 6: Predictions for the photon-jet invariant mass distributions at LO,
NLO, and NNLO. The predictions are compared to data from the ATLAS measurement
ATLAS:2017xqp .

While at NLO the theory prediction fails to describe the shape of the data,
including NNLO corrections a very good agreement with the data in shape and
normalisation is observed. It is only in the second bin that the data within
its uncertainties does not overlap with the NNLO scale uncertainty band.
However, this deviation remains below two standard deviations.

As for the $p_{T}^{\gamma}$ distribution, the contribution from fragmentation
processes to the cross section is small. It amounts to $5\%$ in the first bin
and further decreases for larger transverse momenta.

Figure 6 shows the invariant mass distribution of the photon-jet system. In
the measurement of this distribution additional constraints on
$|\cos\theta^{*}|$ and the sum of the photon pseudorapidity and jet rapidity
are imposed. The NNLO correction reduces the cross section by between $-7\%$
and $-11\%$ compared to NLO and shifts the cross section to the lower edge of
the NLO uncertainty band over the entire invariant mass spectrum. The impact
of the NNLO corrections is most prominent in the region around $m_{\gamma{\rm
jet}}\approx 1000\,\mathrm{GeV}$. As far as scale uncertainties are concerned,
a strong improvement going from NLO to NNLO is observed. While the scale
uncertainty at NLO is 23% in the low mass regime and increases up to 27% in
the tail of the distribution, the maximal scale uncertainty at NNLO is
$(+1,-8)\%$ for $m_{\gamma{\rm jet}}\approx 1000\,\mathrm{GeV}$ and further
decreases for both smaller and larger invariant masses.

Figure 7: Predictions for the photon-plus-jet cross section as a function of
the azimuthal angular separation between the photon and the jet at LO, NLO,
and NNLO. The predictions are compared to data from the ATLAS measurement
ATLAS:2017xqp .

As in the photon transverse momentum spectrum, the NNLO distribution
underestimates the data in the low mass region by 6–8%. In this regime data
and theory predictions are not fully consistent. For larger masses we see
agreement between theory and data within the increasing experimental errors.
We also observe that the shape of the measured distribution is better
described at NNLO compared to NLO. The observation that the measured
distribution in the low mass region as well as in the low to mid
$p_{T}^{\gamma}$ regime falls systematically above the NNLO predictions could
point to an underestimation of the NNPDF4.0 PDF set in this region of phase
space.

The angular separation distribution of the leading jet and the photon is shown
in Figure 7. At LO the photon is recoiling against a single parton, forming
the jet. In this back-to-back configuration $\Delta\phi=\pi$ holds. Smaller
angular separations are accessible only from NLO onwards. The scale
uncertainty at NLO in this region increases with smaller $\Delta\phi$ from 44%
up to 55%. In the back-to-back bin the NLO scale uncertainty is 13%. Including
NNLO corrections reduces the scale uncertainty in this bin to $(+3,-6)\%$. In
the region $3/5\pi<\Delta\phi<9/10\pi$ the scale uncertainty at NNLO amounts
to $(19-28)\%$. The lowest $\Delta\phi$ bin is effectively populated only at
NNLO, resulting in a large NNLO-to-NLO K-factor of almost four and a large
scale uncertainty of $(+33,-22)\%$. NNLO corrections are necessary to describe
the data accurately beyond the back-to-back configuration. Over the complete
$\Delta\phi$ spectrum data and predictions are consistent within their
respective uncertainties.

The contribution from fragmentation processes to the NNLO cross section is
largest in the back-to-back bin where it yields 5 % of the total cross
section. Its relative contribution decreases for smaller azimuthal
separations.

The ATLAS study also provides data for the cross section as a function of
$\cos\theta^{*}\equiv\tanh(\Delta y/2)$, where $\Delta y$ is the rapidity
difference of the jet and the photon. $\theta^{*}$ corresponds to the
scattering angle in the center of mass frame of the underlying $2\to 2$
scattering event. The distribution in $|\cos\theta^{*}|$ is shown in Figure 8.
For the measurement of this distribution the additional cuts (19) on the
invariant mass of the photon-jet system and the sum of jet rapidity and photon
pseudorapidity are imposed. As for the invariant mass distribution, the NNLO
corrections are negative and reduce the NLO cross section by $-6\%$ for small
values of $|\cos\theta^{*}|$ and by roughly $-10\%$ for larger values. The
scale uncertainty at NLO ranges from 15% to more than 50% for
$|\cos\theta^{*}|$ values close to unity. The inclusion of NNLO corrections
yields a strong reduction of scale uncertainty. For small values of
$|\cos\theta^{*}|$ the NNLO scale band is $(+1,-4)\%$ and it moderately
increases to $(+2,-9)\%$ at large $|\cos\theta^{*}|$.

Figure 8: Predictions for the photon-plus-jet cross section differential in
$|\cos\theta^{*}|$ at LO, NLO, and NNLO. The predictions are compared to data
from the ATLAS measurement ATLAS:2017xqp .

Comparing the NNLO prediction to data, we observe good agreement in most bins,
except for the last two for which data are provided. In these two bins the
NNLO prediction falls below the data by approximately 10%. In ATLAS:2017xqp
the $|\cos\theta^{*}|$ distribution was discussed as a candidate for providing
sensitivity on the photon fragmentation contribution in particular at high
values of $|\cos\theta^{*}|$. As discussed in more detail in Section 5.2
below, we indeed observe an increase of the fragmentation contribution to the
full NNLO cross section from 1.5% in the first bin to almost 10% in the region
$|\cos\theta^{*}|\approx 1$. Therefore, the disagreement between theory and
data in the last two bins could point to an underestimation of the
fragmentation contribution by the BFGII set at large photon momentum
fractions.

##  5 Comparison of Different Isolation Prescriptions

In this section we investigate the impact of different isolation prescriptions
on the distributions of the photon-plus-jet cross section. For this comparison
we use the set-up of the 13 $\mathrm{TeV}$ ATLAS study ATLAS:2017xqp  and only
modify the photon isolation.

Our comparison includes two calculations with a fixed cone isolation. As the
default isolation the ATLAS isolation with parameters

$R=0.4,\quad\varepsilon=0.0042,\quad E_{T}^{\rm
thres}=10\,\mathrm{GeV}\qquad\mbox{(default cone)}$ (20)

is used. To investigate the change in magnitude and in composition (direct vs.
fragmentation contribution) of the cross section with increasing maximal
hadronic energy inside the cone $E_{T}^{\rm max}$, we consider a loose fixed
cone isolation with parameters

$R=0.4,\quad\varepsilon=0.0042,\quad E_{T}^{\rm
thres}=50\,\mathrm{GeV}\qquad\mbox{(loose cone)}\,.$ (21)

Previous NNLO calculations for isolated photon Campbell_2017 ; Chen_2020  and
photon-plus-jet production Chen_2020 ; Campbell:2017dqk  used idealised photon
isolations, i.e. smooth cone or hybrid cone prescriptions. In the following
comparison we confront a hybrid isolation with parameters

$R=0.4,\;\varepsilon=0.0042,\;E_{T}^{\rm
thres}=10\,\mathrm{GeV},\;R_{d}=0.1,\;\varepsilon_{d}=0.1,\;n=2\quad\mbox{(hybrid)}$
(22)

with the default fixed cone isolation. The parameters of the inner cone are
tuned to mimic the ATLAS fixed cone isolation Chen_2020 . Therefore, one
objective of the following comparison is to reveal to which extent the hybrid
isolation faithfully reproduce the experimental isolation prescription at NNLO
accuracy.

In the democratic clustering isolation approach Glover:1993xc ; Hall:2018jub
photons and partons are clustered together. This democratic treatment of
photons and partons can be particularly useful for calculations of electroweak
corrections were additional photons are emitted. To test the impact of
democratic clustering on QCD corrections we apply an anti-$k_{T}$ algorithm
with cone radius $R_{{\rm jet}}=0.4$ and a $z_{\rm cut}=0.926$. The criterion
$z_{\rm em}>z_{\rm cut}$ with $z_{\mathrm{em}}$ denoting the electromagnetic
energy fraction (4) can be reformulated as a criterion on the maximal hadronic
energy inside the cone of the democratic jet. The maximal integrated hadronic
energy inside this cone may not exceed

$E_{T}^{\rm max}=\frac{1-z_{\rm cut}}{z_{\rm cut}}\,p_{T}^{\gamma}\,.$ (23)

The value $z_{\rm cut}=0.926$ is chosen such that at
$p_{T}^{\gamma}=125\,\mathrm{GeV}$ the maximal hadronic energy is
$10\,\mathrm{GeV}$, which coincides with the value of $E_{T}^{\rm thres}$ of
the ATLAS isolation criterion.

In addition to these isolated photon-plus-jet calculations, we also present
results for the photon-plus-jet cross section without imposing any photon
isolation. Even if this inclusive set-up may not be feasible in an
experimental measurement, it can provide valuable insights on the overall
impact of applying an isolation criterion to the photon.

The comparison of the isolation prescriptions is performed in two steps. In
Section 5.1 we examine the impact of the isolations on the magnitude of the
cross sections. A detailed analysis of the composition of the cross sections
in terms of direct and fragmentation components follows in Section 5.2.

###  5.1 Comparison of the NNLO Cross Sections

Figure 9: Predictions for the photon transverse momentum (left) and jet
transverse momentum distributions (right) with different isolation
prescriptions.

For the comparison of the isolation prescriptions we use our default scale
choices $\mu_{R}=\mu_{F}=p_{T}^{\gamma}$ and $\mu_{A}=m_{\rm cone}$. The
predictions with hybrid isolation do not contain a contribution from
fragmentation processes and therefore they do not depend on a fragmentation
scale. In this case the scale uncertainty is estimated by a seven-point scale
variation of the renormalisation and factorisation scales. It is noted that
for the hybrid isolation an additional uncertainty arises from the choice of
the isolation parameters $(R_{d},\,\varepsilon_{d},\,n)$ in (22). We do not
quantify this uncertainty here but stress that it is not captured by the scale
uncertainty. For the inclusive predictions without photon isolation the
standard choice for the fragmentation scale cannot be applied as there is no
isolation cone. For this inclusive set-up we use $\mu_{A}=p_{T}^{\gamma}$,
since the fragmentation process is in this case inclusive with respect to all
extra radiation up to the characteristic scale $p_{T}^{\gamma}$ of the final
state.

The different predictions for the $p_{T}^{\gamma}$ and $p_{T}^{\rm jet}$
distributions of the photon-plus-jet cross section are compared in Figure 9.
In the lower panels we show the ratio of the individual predictions with
respect to the default ATLAS isolation.

Comparing the photon transverse momentum distributions, we observe a
difference between the hybrid and default isolation of roughly 4% for small
$p_{T}^{\gamma}$. For larger values this difference is reduced and at
$p_{T}^{\gamma}\approx 1000\,\mathrm{GeV}$ the two predictions largely
overlap. A similar behaviour is found for the loose isolation and the
inclusive photon-plus-jet cross section. At small transverse momenta a 10%
(24%) difference compared to the default isolation is observed for the
prediction with a loose (without) isolation. With increasing $p_{T}^{\gamma}$
the distributions approach each other. The loose isolation cross section fully
coincides with the default cross section for $p_{T}^{\gamma}=2\,\mathrm{TeV}$,
while the inclusive remains 6% larger. This behaviour indicates that photons
at large transverse momenta are likely to be accompanied by only little
hadronic energy so that the impact of the photon isolation is only moderate.

The slope of the democratic clustering distribution is not as steep as the
slope of the default distribution. This behaviour can be understood from the
functional form of the effective maximal hadronic energy inside the democratic
jet given in (23). At $p_{T}^{\gamma}=125\,\mathrm{GeV}$ the maximal hadronic
energy for democratic clustering is $E_{T}^{\rm max}=10\,\mathrm{GeV}$ and
therefore similar to the corresponding value for the default isolation
$E_{T}^{\rm max}=10.5\,\mathrm{GeV}$. However, in the democratic clustering
approach it increases more strongly with increasing $p_{T}^{\gamma}$ compared
to the default isolation. At $p_{T}^{\gamma}=1000\,\mathrm{GeV}$ the maximal
hadronic energy is $E_{T}^{\rm max}=80\,\mathrm{GeV}$ for democratic
clustering, while for the default criterion it is $E_{T}^{\rm
max}=14.2\,\mathrm{GeV}$. Therefore, in the high $p_{T}^{\gamma}$ regime the
democratic clustering cross section does even exceed the cross section
obtained with a loose isolation. An important observation is that the region
of largest discrepancy between the isolation prescription coincides with the
region where the contribution from fragmentation processes is strongest. This
will be further analysed in Section 5.2, where the composition of the cross
sections with different isolation prescriptions is discussed.

The difference between the hybrid isolation and the default isolation in the
$p_{T}^{\rm jet}$ distribution is also most prominent at low transverse
momenta. In the first bin the hybrid isolation overshoots the default
isolation by 10% while in the second bin it falls below the default isolation
by 5%. For $p_{T}^{\rm jet}>170\,\mathrm{GeV}$ the shapes of the two
distributions coincide and the hybrid isolation exceeds the default isolation
by only a few percent. The distribution obtained with a loose isolation falls
above the default isolation over the entire range of jet transverse momenta.
The largest difference between the two isolations is observed in the regime
$130\,\mathrm{GeV}<p_{T}^{\rm jet}<300\,\mathrm{GeV}$ where the corresponding
ratio reaches almost 1.13.

In the first bin of the $p_{T}^{\rm jet}$ distribution the inclusive (no
isolation) photon-plus-jet cross section is smaller than the cross section
obtained with the default isolation. This peculiar behaviour stems from the
Sudakov shoulder Catani:1997xc  at $p_{T}^{\rm jet}=125\,\mathrm{GeV}$ causing
a poor convergence of the perturbative expansion in this regime. Towards
larger jet transverse momenta a strong increase in the ratio is observed. As
discussed in Section 4.4 this regime is dominated by events with two hard
recoiling jets accompanied by a relatively soft photon (still fulfilling the
$p_{T}^{\gamma}$ cut). A large contribution to this configuration stems from
fragmentation processes, where a high-$p_{T}$ jet fragments into a photon and
passes a moderate momentum fraction of its original momentum to the photon.
Such fragmentation processes in this kinematic configuration are almost fully
vetoed by all isolation criteria, since the photon is accompanied by a large
amount of hadronic energy. The strongly increasing scale uncertainty in the
tail of the distribution is caused by the particular scale choice
$\mu_{R}=\mu_{F}=\mu_{A}=p_{T}^{\gamma}$, yielding large logarithms of the
ratio $p_{T}^{\rm jet}/p_{T}^{\gamma}$ in this regime.

An increasing ratio between the democratic clustering isolation and the
default isolation can also be seen in the $p_{T}^{\rm jet}$ distribution. In
the lowest bin the cross section obtained with democratic clustering is 7%
smaller than the default cross section while at high $p_{T}^{\rm jet}$ the
democratic clustering cross section exceeds the default cross section by 10%.

Figure 10: Predictions for the photon-jet invariant mass (left) and the
leading jet rapidity distribution (right) using different isolation
prescriptions.

Figure 10 shows the invariant mass distribution of the photon–jet system as
well as the rapidity distribution of the leading jet. For the invariant mass
distribution the additional cuts (19) are imposed. Comparing the invariant
mass distribution of the default and the hybrid isolation, we observe the
largest difference in the low mass region. At $m_{\gamma{\rm
jet}}=500\,\mathrm{GeV}$ the hybrid isolation overestimates the ATLAS
isolation by 4%. The difference between the two isolation prescriptions
vanishes at very high invariant masses. For $m_{\gamma{\rm
jet}}=3000\,\mathrm{GeV}$ we observe only a 1% difference and the scale
uncertainty bands of both predictions largely overlap.

Figure 11: Predictions for the photon-plus-jet cross section differential in
$|\cos\theta^{*}|$ (left) and in the azimuthal angular separation between the
photon and the jet (right) using different isolation prescriptions.

Comparing the loose and the default isolation, the largest difference is found
in the low mass regime where the loose isolation prediction exceeds the
default isolation by 11%. With increasing masses the corresponding ratio
decreases and it amounts to only 1.02 in the highest mass bin.

In the mass regime around $m_{\gamma{\rm jet}}\approx 600\,\mathrm{GeV}$ a
strong increase of the inclusive photon-plus-jet distribution compared to the
default distribution is seen. The relative difference of the two distributions
reaches its maximum around 1000 $\mathrm{GeV}$. For larger masses the
corresponding ratio moderately decreases to a value of 1.5.

The distribution obtained with a democratic clustering isolation shows a
slightly different slope compared to the default distribution and the
corresponding ratio increases from 1.01 to roughly 1.05 in the very high mass
regime. This indicates that the large $m_{\gamma{\rm jet}}$ regime is
dominated by high-$p_{T}$ photons for which the two isolation criteria
deviate.

The shapes of the rapidity distributions of the leading jet coincide for the
different isolation prescriptions, yielding constant ratios over the whole
rapidity range. This is in line with the expectation that the entire rapidity
range is dominated by low-$p_{T}$ jets events. The corresponding ratio can be
inferred from the ratios obtained for the first few bins of the jet transverse
momentum distribution. The hybrid isolation systematically falls above the
default isolation by about $4\%$, the loose isolation by about $9\%$ and the
inclusive photon-plus-jet cross section by about $20\%$. The democratic
isolation prescription matches the default isolation not only in shape but
also in magnitude, as is expected given that the parameters were tuned to
match at low $p_{T}^{\gamma}$ where the cross section is largest. Both
isolations yield very similar distributions. The fragmentation contribution to
the different distributions is constant over the considered rapidity range.

The $|\cos\theta^{*}|$ distribution and the $\Delta\phi$ distribution are
shown in Figure 11. $\theta^{*}$ corresponds to the polar scattering angle of
the underlying $2\to 2$ scattering event. Due to a different functional
dependence on $\cos\theta^{*}$ of the underlying matrix elements, the
contribution from fragmentation is most prominent in the high
$|\cos\theta^{*}|$ regime (see also Section 5.2). The largest difference
between the hybrid and the default isolation is observed in this regime, where
it amounts to almost 13%. The loose isolation distribution also deviates
strongest from the default isolation for large $|\cos\theta^{*}|$. The
corresponding ratio increases from 1.06 at small $|\cos\theta^{*}|$ to 1.17 at
$|\cos\theta^{*}|=1$. In contrast to the loose and hybrid isolation, the
democratic clustering distribution agrees with the default isolation
distribution over the full range of $|\cos\theta^{*}|$ within the respective
scale uncertainty bands. This indicates that the impact of events with very
high-$p_{T}$ photons to the $|\cos\theta^{*}|$ distribution is small. The
inclusive photon-plus-jet cross section exceeds all isolated photon-plus-jet
cross sections. The largest difference between the inclusive distribution and
the default distribution is observed at $|\cos\theta^{*}|=0.65$ where it
amounts to roughly 55%. The corresponding ratio decreases for larger and
smaller values of $|\cos\theta^{*}|$.

The different $\Delta\phi$ distributions almost fully coincide for
$\pi/2<\Delta\phi<3\pi/5$. In this regime the photon is recoiling against at
least two additional partons. Therefore, it is likely to be well separated
from additional hadronic energy and the isolation prescription does not
strongly impact the cross section. This is in line with the observation that
fragmentation processes yield only a negligible contribution to the cross
section in this kinematic range. The difference between the different
isolations increases with increasing $\Delta\phi$. The largest discrepancy is
observed in the back-to-back bin. In this bin the inclusive distribution is
25% larger than the default distribution and the loose isolation distribution
overshoots the default isolation by 9%. The distribution obtained with
democratic clustering matches the default distribution over the whole range of
$\Delta\phi$, which shows that the impact of photons with large transverse
momenta to this distribution is very small. The hybrid isolation overestimates
the default isolation for $7/10\pi<\Delta\phi<\pi$ by approximately 4%. Since
the fragmentation contribution to the cross section increases with
$\Delta\phi$, this confirms observations in other distributions that the
difference between the hybrid and default isolation is largest in regions
where also the fragmentation contribution is largest.

A particularly remarkable feature of all distributions is the close agreement
between democratic isolation and default cone-based isolation, which is
observed everywhere except for the high-$p_{T}^{\gamma}$ tail of the photon
transverse momentum distribution. This agreement is most likely a consequence
of the fact that the anti-$k_{T}$ algorithm Cacciari:2008gp  that is used for
the clustering in the democratic isolation approach results in an almost
perfectly cone-shaped capture area of radius $R_{\rm jet}$ Cacciari:2008gn
(which is chosen identical to the isolation cone radius $R$ here), and that
the $z_{{\rm cut}}$ applied subsequently on the photon jet is determined to
reproduce $E_{T}^{\rm max}$ of the default isolation at the lower end of the
$p_{T}^{\gamma}$ distribution, where the cross section is largest. With
parameters chosen appropriately, the effect of the democratic isolation
procedure based on the anti-$k_{T}$ algorithm is thus very similar to a fixed
cone isolation.

###  5.2 Decomposition of the Cross Sections

Figure 12: Decomposition of the cross section differential in the photon
transverse momentum. Results obtained with the default isolation, a loose
isolation, democratic clustering and without isolation are shown.

In this section we investigate the impact of the isolation prescription on the
composition of the photon-plus-jet cross section. As the hybrid isolation only
contains a contribution from direct photon production processes, this
isolation is not considered in the following.

In Figure 12 the photon transverse momentum distributions for the default
isolation, the loose isolation, democratic clustering isolation, and the
inclusive set-up are shown. Besides the decomposition of the full NNLO cross
section (red) into the direct (yellow) and fragmentation contribution (green)
according to (7), the LO and NLO predictions are displayed. In the LO
approximation the photon is in a back-to-back configuration with one
additional parton. Therefore, it is well separated from any hadronic energy
and the isolation prescription does not affect the cross section at this
order. This does no longer hold from NLO onward, where additional real
radiation processes as well as fragmentation processes start to contribute.

Figure 13: Decomposition of the cross section differential in the jet
transverse momentum. Results obtained with the default isolation, a loose
isolation, democratic clustering and without isolation are shown.

For the default isolation, the loose isolation and the inclusive set-up the
contribution from fragmentation processes to the full NNLO cross section
declines with increasing $p_{T}^{\gamma}$. Moreover, the looser the isolation
parameters are the larger is the fragmentation contribution to the photon
transverse momentum distribution. For the default isolation the impact of the
fragmentation contribution to the full NNLO cross section is 5% in the lowest
bin and beyond $p_{T}^{\gamma}=600\,\mathrm{GeV}$ it decreases to less than
1%.

With a loose photon isolation a higher sensitivity on the fragmentation
processes is obtained. At small photon transverse momenta 20% of the full NNLO
cross section stems from these processes. Without photon isolation, direct
photons and fragmentation each yield roughly 50% of the cross section at small
$p_{T}^{\gamma}$ and in the high $p_{T}^{\gamma}$ regime the fragmentation
contribution still amounts to 25% of the cross section. For the democratic
clustering isolation the fragmentation contribution moderately increases with
increasing $p_{T}^{\gamma}$ from 5% to 8%. This is in line with the
observation in Figure 9 where the strongest discrepancy to the default
isolation was found in the high $p_{T}^{\gamma}$ regime.

Figure 14: Decomposition of the cross section differential in the photon-jet
invariant mass. Results obtained with the default isolation, a loose
isolation, democratic clustering and without isolation are shown.

Figure 13 shows the decomposition of the leading jet transverse momentum
distribution for the different isolations. Comparing the default and loose
isolation, an increase in the sensitivity on fragmentation processes with
looser isolation parameters is observed. The predictions obtained with the
default isolation contain a relative fragmentation contribution of $5\%$ for
low $p_{T}^{\rm jet}$ whereas for the loose isolation the fragmentation
contribution is $20\%$ in this regime. As for the $p_{T}^{\gamma}$
distribution, the impact from fragmentation processes decreases with
increasing transverse momenta.

In the democratic clustering approach the relative contributions from direct
photons and photon fragmentation remain almost constant over the complete
kinematic range. For this isolation roughly 4% of the photons originate from
fragmentation processes.

For the inclusive set-up the high $p_{T}^{\rm jet}$ region is completely
dominated by fragmentation processes. Here the direct contribution only
amounts to 15% of the NNLO cross section. This strong increase of
fragmentation processes causes the large difference between the inclusive and
isolated photon cross sections displayed in Figure 9. At large jet transverse
momenta, events with two hard recoiling jets accompanied by a relatively soft
photon yield the largest contribution to the cross section. In this
configuration the allowed range of momentum fractions which one of the jet
passes to the photon during fragmentation increases with increasing leading
jet transverse momentum. The accessible momentum fraction range for the
fragmentation of a 1 $\mathrm{TeV}$ jet is $z\in[0.125,1]$, while for a jet
with 150 GeV transverse momentum it is only $z\in[0.83,1]$. The fragmentation
functions are strongly increasing towards lower momentum fractions. Therefore,
in the majority of events in the tail of the inclusive $p_{T}^{\rm jet}$
distribution the photon is accompanied by a large amount of hadronic energy.

The breakdown of the invariant mass distribution of the photon–jet system is
shown in Figure 14. The composition of the cross section as a function of the
invariant mass is very similar to the composition of the $p_{T}^{\gamma}$
distributions, reflecting the functional dependence of $m_{\gamma{\rm jet}}$
on $p_{T}^{\gamma}$. For the fixed cone isolations, the contribution from
fragmentation is largest for small masses where it amounts to 20% for the
loose isolation and 4% for the default isolation. Towards larger masses the
fragmentation contribution declines strongly. The invariant mass distribution
obtained with democratic clustering contains an increasing contribution from
fragmentation processes for larger masses. For $m_{\gamma{\rm jet}}\approx
500\,\mathrm{GeV}$ the contribution from fragmentation to the NNLO cross
section is 6% and it increases to 9% at $m_{\gamma{\rm jet}}\approx
3\,\mathrm{TeV}$.

The fragmentation contribution dominates the non-isolated NNLO cross section
over the entire range of masses. Its relative size increases from 56% for low
masses to 66% in the tail of the distribution. The invariant mass distribution
also shows that the scale uncertainties of the fragmentation and the direct
contribution largely compensate each other yielding only a very small scale
band of the full NNLO cross section.

Comparing Figure 14 to the invariant mass distribution in Figure 10, we
observe a similar correlation between the sensitivity on fragmentation
processes and the sensitivity on the isolation prescription, i.e. the regions
in which the isolation prescriptions deviate are the regions with the largest
fragmentation contribution.

The difference between photon pseudorapidity and jet rapidity is related to
the scattering angle in the parton-parton centre-of-momentum frame, with large
rapidity difference corresponding to small scattering angles. Predictions for
the cross section as a function of the rapidity difference are shown in Figure
15. For all isolations the cross section is falling with increasing
$|\eta_{\gamma}-y_{\rm jet}|$. Moreover, for very large rapidity differences
the NNLO predictions fall below the scale uncertainty band of the NLO
predictions indicating a poor perturbative convergence of the cross section in
this kinematic regime.

The relative size of the fragmentation contribution increases towards larger
rapidity differences for all considered isolations. At $|\eta_{\gamma}-y_{\rm
jet}|=4$, fragmentation processes amount to 11% of the full NNLO cross section
for the default isolation. For even larger rapidity differences a strong
decline of the direct contribution is observed, yielding even larger relative
contributions from fragmentation processes. In the democratic clustering
approach we observe a very similar behaviour of the cross section. Using this
isolation the relative size of the fragmentation contribution also increases
towards very unbalanced configurations and at $|\eta_{\gamma}-y_{\rm jet}|=4$
fragmentation processes amount to 16% of the NNLO cross section.

Figure 15: Decomposition of the cross section differential in the rapidity
difference between the photon and the jet. Results obtained with the default
isolation, a loose isolation, democratic clustering and without isolation are
shown.

No qualitative change is found for the loose isolation prescription. The
impact of the fragmentation contribution grows from 10% for small rapidity
differences to 44% at $|\eta_{\gamma}-y_{\rm jet}|=4$. We note that it is only
in these very unbalanced configurations that the fragmentation contribution is
roughly as large as the direct contribution for the loose isolation.

For the inclusive set-up the direct contribution dominates for
$|\eta_{\gamma}-y_{\rm jet}|<1.4$. For larger rapidity differences
fragmentation processes dominate the NNLO cross section. For rapidity
differences larger than four the contribution from direct photons is
negligible.

Figure 16: Decomposition of the cross section differential in the azimuthal
separation between the photon and the jet. Results obtained with the default
isolation, a loose isolation, democratic clustering and without isolation are
shown.

The cross sections as a function of the angular separation of the photon and
the jet are shown in Figure 16. For all isolation set-ups the relative size of
the fragmentation contribution grows with increasing $\Delta\phi$ and it is
largest in the back-to-back bin. Figure 11 shows that the default isolation
and the democratic clustering yield very similar $\Delta\phi$ distributions.
This observation also holds for the composition of the cross section. For both
isolations the relative contribution from fragmentation increases from 0.5 %
at $\pi/2<\Delta\phi<3\pi/5$ to roughly 5% in the back-to-back bin.

Direct photon production also dominates the $\Delta\phi$ distribution obtained
with a loose isolation. For small $\Delta\phi$ direct photons contribute
almost exclusively, while in back-to-back configurations 20% of the cross
section is given by fragmentation processes.

Figure 17: Decomposition of the cross section differential in
$|\cos\theta^{*}|$. Results obtained with the default isolation, a loose
isolation, democratic clustering and without isolation are shown.

Dropping the isolation requirement, the fragmentation contribution increases
strongly. However, also with this set-up the lowest $\Delta\phi$ bin is
largely dominated by direct photons. The fragmentation contribution in this
bin is only 6%. In the back-to-back bin the fragmentation contribution is
roughly 50%. The large suppression of the fragmentation contribution in the
small $\Delta\phi$-regime is due to an insufficient number of real radiation
partons in fragmentation processes to effectively populate this kinematic
regime. The large NNLO-to-NLO K-factor for the full cross section in the first
bin shows that at least three additional partons besides the photon are needed
to effectively populate this bin. In our predictions for the fragmentation
contribution only processes with up to two additional partons are included.
Therefore, it is expected that including higher order fragmentation processes
would yield a strong increase in the fragmentation contribution for small
$\Delta\phi$.

The observable $|\cos\theta^{*}|$ was discussed by ATLAS ATLAS:2017xqp  as a
potential candidate for providing sensitivity on fragmentation processes in
particular at large values of $|\cos\theta^{*}|$. The different isolated cross
sections as well as the non-isolated cross section differential in
$|\cos\theta^{*}|$ are shown in Figure 17. As expected from the different
dependence of the underlying Born matrix elements for direct and fragmentation
processes on $\cos\theta^{*}$, the relative fragmentation contribution
increases for all set-ups towards larger $|\cos\theta^{*}|$.

For small $|\cos\theta^{*}|$ direct photons dominate the cross section for all
considered isolations. The fragmentation contribution for
$|\cos\theta^{*}|<0.3$ amounts to only 1.5% of the full NNLO cross section
with the default isolation. For democratic clustering 3% of the cross section
originate from fragmentation processes and for the loose isolation 8%. In the
bin $0.7<|\cos\theta^{*}|<0.83$, which is the highest bin for which the ATLAS
measurement provides data, the relative contribution from fragmentation is 6%
for the default isolation, 8% for democratic clustering and 27% for the loose
fixed cone isolation. For even larger values of $|\cos\theta^{*}|$, which have
not been accessed by experimental measurements yet, the relative fragmentation
contribution increases even further.

For the non-isolated cross section direct photon and fragmentation contribute
almost equally for $|\cos\theta^{*}|<0.4$. Towards larger values of
$|\cos\theta^{*}|$ a strong decline of the direct contribution is observed and
at $|\cos\theta^{*}|\approx 1$ fragmentation contributes 90% to the full NNLO
cross section.

##  6 Imbalance between Photon and Jet Transverse Momentum: $z_{\rm rec}$

At large longitudinal momentum transfers, the photon fragmentation functions
are only weakly constrained through experimental measurements at LEP from the
ALEPH Buskulic:1995au  and OPAL Ackerstaff:1997nha  experiments. However, for
(tightly) isolated photon production the fragmentation contribution almost
exclusively stems from events in which a large fraction of transverse momentum
is passed to the photon. To constrain the photon fragmentation functions in
this kinematical region, several observables in electron-positron annihilation
Glover:1993xc , deep inelastic scattering Gehrmann-DeRidder:2006zbx ;
Gehrmann-DeRidder:2006lpc  and at hadron colliders Belghobsi:2009hx ;
Kaufmann:2016nux  have been proposed. These proposed observables derive their
sensitivity on the photon fragmentation functions at large momentum transfer
from either investigating the substructure of a jet containing a highly
energetic photon or from kinematic photon-jet correlations. Having very
precise experimental data and theory predictions which incorporate the
fragmentation processes at NNLO, it can now be envisaged to determine the
photon fragmentation functions in this regime using isolated photon and
photon-plus-jet data from the LHC.

In this section we study the observable $z_{\rm rec}$, which provides
differential sensitivity on the photon fragmentation functions and therefore
could help to constrain the photon fragmentation functions in future studies.
The variable $z_{\rm rec}$ is a measure for the imbalance of the photon and
leading jet transverse momenta, i.e.

$z_{\rm rec}\equiv\frac{p_{T}^{\gamma}}{p_{T}^{\rm jet}}\,.$ (24)

A similar variable $z_{\gamma}$, including a projection of the photon momentum
onto the leading jet axis, has been proposed in Belghobsi:2009hx . For the LO
cross section of direct photon production, the two objects are exactly
balanced in the transverse plane and thus $z_{\rm rec}=1$ holds. From NLO
onwards additional parton radiation can generate configurations for which
$z_{\rm rec}$ is not equal to unity. In this case we can distinguish two
different scenarios. If the additional parton is radiated in the hemisphere of
the photon we have $z_{\rm rec}<1$ while if the additional parton is radiated
in the opposite hemisphere and it is not clustered with the other parton by
the jet algorithm we have $z_{\rm rec}>1$.

For photon fragmentation at Born level, two back-to-back jets (partons) are
radiated from which one subsequently fragments into a photon and passes a
fraction $z$ of its momentum to the photon. This process is described by the
parton-to-photon fragmentation function $D_{p\to\gamma}(z)$. In this case the
imbalance between the transverse momentum of the photon and the leading jet is
given by

$z_{\rm rec}=\frac{p_{T}^{\gamma}}{p_{T}^{\rm jet}}=\frac{z\,p_{T}^{\rm
jet}}{p_{T}^{\rm jet}}=z\,.$ (25)

Consequently, at this order there is a one-to-one correspondence between
$z_{\rm rec}$ and the argument of the fragmentation function $z$. The range of
$z_{\rm rec}$ values which is populated by lowest order fragmentation
processes can be inferred from the corresponding isolation parameters. For a
fixed cone isolation with parameter $(R,\,\varepsilon,\,E_{T}^{\rm thres})$
and assuming that $\varepsilon\ll 1$, we have

$1>z_{\rm rec}>z_{\rm rec}^{\rm min}=\frac{p_{T_{\rm min}}^{\gamma}}{p_{T_{\rm
min}}^{\gamma}+E_{T}^{\rm thres}}\,,$ (26)

where $p_{T_{\rm min}}^{\gamma}$ is the minimal photon transverse momentum.
Beyond lowest order, the one-to-one correspondence between $z_{\rm rec}$ and
the argument of the fragmentation function does no longer hold. Moreover, the
radiation of additional partons extends the range of populated $z_{\rm rec}$
values to $z_{\rm rec}>1$ and $z_{\rm rec}<z_{\rm rec}^{\rm min}$.

Figure 18: Decomposition of the cross section differential in $z_{\rm rec}$.
Results obtained with the default isolation, a loose isolation, democratic
clustering and without isolation are shown.

Results for the $z_{\rm rec}$ distributions of the photon-plus-jet cross
section are shown in Figure 18. For the predictions we used the set-up of the
ATLAS 13 $\mathrm{TeV}$ study ATLAS:2017xqp  and compared the default
isolation, the fixed cone loose isolation, democratic clustering and the non-
isolated cross section. For all set-ups we observe an increase of the NNLO
cross section towards $z_{\rm rec}\approx 1$. Due to the Sudakov shoulder at
$z_{\rm rec}=1$, the distributions show a poor perturbative convergence in the
near vicinity of $z_{\rm rec}=1$. Considering the NNLO-to-NLO K-factors of the
$z_{\rm rec}$ distributions, the region in which the fixed-order calculation
fails to give a reliable description can be estimated to be $z_{\rm
rec}\in[0.93,1.06]$. The isolated distributions are dominated by direct
photons over most of the kinematic range. In the region $z_{\rm rec}<1$ the
fragmentation contribution is increasing with $z_{\rm rec}$ only below the
value $z_{\rm rec}^{\rm min}$ in (26), corresponding to a kinematical region
that requires the fragmenting photon cluster to be accompanied by at least two
other partons and that is described only at NNLO and beyond. For the default
isolation and for democratic clustering we have $z_{\rm rec}^{\rm min}=0.926$
and for the loose isolation $z_{\rm rec}^{\rm min}=0.714$. At $z_{\rm
rec}^{\rm min}$ the fragmentation contribution drops, resulting in a kink of
the full NNLO cross section, which is most prominent for the loose isolation.
For $z_{\rm rec}^{\rm min}<z_{\rm rec}<1$ the contribution from fragmentation
is very small for the default isolation and for democratic clustering it even
turns negative. Some modest amount of fragmentation contribution remains for
the loose isolation prescription.

Figure 19: Predictions for the photon-plus-jet cross section differential in
$z_{\rm rec}$ using different isolation prescription.

A completely different picture emerges if no photon isolation is applied. In
this case, the direct production processes dominate the $z_{\rm rec}$
distribution only for $z_{\rm rec}>1$. We observe a strong increase in the
fragmentation contribution for $z_{\rm rec}<1$. Already for $z_{\rm
rec}<0.89$, fragmentation processes dominate over direct photon production and
for even smaller values of $z_{\rm rec}$ fragmentation processes contribute
almost exclusively. Again, the immediate vicinity of $z_{\rm rec}=1$ can not
be described by fixed-order perturbation theory due to the Sudakov shoulder
behaviour. In contrast to the $z_{\rm rec}$ distributions for different photon
isolation prescriptions, no further perturbative instability is observed
around a $z_{\rm rec}^{\rm min}<1$ value.

Figure 19 compares the $z_{\rm rec}$ distributions at NNLO obtained for the
different photon isolation prescriptions, also including hybrid isolation.
Substantial deviations outside the theory uncertainties from the default ATLAS
isolation are observed only if no isolation is applied. While democratic
clustering yields a distribution almost identical to the default isolation,
loose and hybrid isolation predict a uniform but not significant enhancement
of the distribution for $z_{\rm rec}<1$. The substantial enhancement of the
non-isolated $z_{\rm rec}$ distribution for $z_{\rm rec}<1$ is entirely due to
photon fragmentation processes. This observation strongly suggests that a
determination of the photon fragmentation functions at the LHC could
potentially be attempted by a measurement of the $z_{\rm rec}$ distribution in
photon-plus-jet events without any isolation requirements on the photon. The
sensitivity on the fragmentation functions could potentially be enhanced
further by taking the difference or a ratio between isolation prescriptions
and/or applying additional fiducial selection cuts, e.g.
$|\cos\theta^{*}|\gtrsim 0.5$ or $\Delta\phi\gtrsim\tfrac{3\pi}{4}$, to
enhance the relative contribution from the fragmentation processes.

We are perfectly aware of the substantial experimental challenges that a
measurement of photon production cross sections without any isolation
requirement entails. Its demonstrated sensitivity on the photon fragmentation
functions presents a strong motivation to further investigate the prospects of
such a measurement, and warrants a detailed assessment of potential
experimental challenges.

##  7 Conclusions

In this paper, we used the antenna subtraction method to compute the NNLO QCD
corrections to isolated photon and photon-plus-jet production, including
contributions Gehrmann:2022cih  from collinear photon radiation off partons
and parton-to-photon fragmentation. Previous calculations at this order
Campbell:2017dqk ; Campbell_2017 ; Chen_2020  discarded these contributions by
applying an idealised photon isolation based on a dynamical isolation cone
Frixione:1998jh . Our newly derived results enable us for the first time to
reproduce the photon isolation procedure used in the experimental
measurements, which is based on a fixed-size cone around the photon direction,
admitting a finite amount of hadronic energy. Our results are implemented in
the NNLO  JET parton-level event generator. A detailed comparison with
experimental results from ATLAS, CMS and ALICE demonstrated the impact of the
NNLO corrections for precision phenomenology.

The mass factorisation scale in the photon fragmentation functions separates
direct and fragmentation contributions; it reflects up to which scale an
observable is inclusive in extra radiation accompanying the photon. We argue
that in the case of isolated photons, this scale is related to the maximal
invariant mass of photons and partons inside the isolation cone, which is
typically much lower than the photon transverse momentum. Choosing a larger
mass factorisation scale in the photon fragmentation functions would imply to
cut out a substantial amount of out-of-cone parton radiation from the direct
process at fixed order, which is added back in resummed (and thus attenuated)
form through the fragmentation functions, thereby leading to an erroneous
description of out-of-cone dynamics. With the invariant mass scale choice in
the photon fragmentation functions, the fragmentation contributions are small
and vanish with increasing photon transverse momentum. The smallness of the
factorisation scale and the overall small impact of the fragmentation
contributions justifies a power counting of $\alpha$ for the fragmentation
functions, which is used throughout in our work.

The newly derived NNLO corrections allow us to compute predictions for photon-
plus-jet production without any photon isolation and to compare different
isolation prescriptions. The application of the default ATLAS photon isolation
leads to a decrease of the non-isolated cross section by only 30%. Results
obtained with a dynamical cone are close in normalisation to the default
fixed-cone predictions, but deviate in the kinematical shape especially at low
photon transverse momenta. Loosening of the isolation parameters increases the
cross sections, with the largest effects observed at low photon or jet
transverse momentum.

The democratic clustering procedure for photon isolation Glover:1993xc ;
GehrmannDeRidder:1997gf  treats photons as any other partons in the jet
clustering, identifying the photons only after the jet clustering by labelling
all jets containing more than a predefined large fraction of electromagnetic
energy as isolated photons. This isolation procedure will be particularly
advantageous when electroweak corrections to photon observables are taken into
account, since it eliminates all ambiguities relating to a discrimination in
kinematical handling between identified and non-identified photons. It has
however been proven impractical to be implemented in experimental analyses so
far. We demonstrate that by using democratic clustering in the anti-$k_{T}$
algorithm with a cone size identical to the fixed cone isolation and properly
adjusted energy fraction threshold, the predicted cross sections are nearly
identical to those obtained for fixed cone isolation. These results may help
to enable measurements applying democratic clustering especially for
electroweak precision observables.

We introduce the observable $z_{\rm rec}$ derived from photon-plus-jet final
states, which allows us to identify kinematical regions that are potentially
sensitive on photon fragmentation contributions and the form of the
fragmentation functions. The highest sensitivity to parton-to-photon
fragmentation functions is observed if no photon isolation is applied, and we
discuss possible hadron collider measurements of these functions.

Our newly derived results eliminate the systematic uncertainty resulting from
the mismatch of photon isolation prescription between experiment and theory
for isolated photon final states at hadron colliders. They will thus enable a
broad range of precision phenomenology studies with these observables,
including electroweak precision studies, constraints on parton distributions,
and the determination of photon fragmentation functions.

## Acknowledgements

The authors thank Juan Cruz-Martinez, James Currie, Rhorry Gauld, Aude
Gehrmann-De Ridder, Imre Majer, Matteo Marcoli, Jonathan Mo, Thomas Morgan,
Jan Niehues, João Pires, Adrian Rodriguez Garcia, Giovanni Stagnitto, Duncan
Walker, Steven Wells and James Whitehead for useful discussions and their many
contributions to the NNLO  JET code. We would also like to thank Thomas Becher
and Joey Huston for interesting discussions on photon isolation and its
implications. This work has received funding from the Swiss National Science
Foundation (SNF) under contract 200020-204200, from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme grant agreement 101019620 (ERC Advanced Grant TOPUP), from the UK
Science and Technology Facilities Council (STFC) through grant ST/T001011/1
and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under grant 396021762-TRR 257.

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