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# Searching for axion-like particles with data scouting at ATLAS and CMS

Simon Knapen<EMAIL_ADDRESS>CERN, Theoretical Physics Department, Geneva,
Switzerland  Berkeley Center for Theoretical Physics, Department of Physics,
University of California, Berkeley, CA 94720, USA  Theoretical Physics Group,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  Soubhik Kumar
<EMAIL_ADDRESS>Berkeley Center for Theoretical Physics, Department of
Physics, University of California, Berkeley, CA 94720, USA  Theoretical
Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Diego Redigolo<EMAIL_ADDRESS>CERN, Theoretical Physics Department,
Geneva, Switzerland  INFN, Sezione di Firenze Via G. Sansone 1, 50019 Sesto
Fiorentino, Italy

(August 27, 2024)

###### Abstract

We investigate the physics case for a dedicated trigger-level analysis for
very low mass diphoton resonances at ATLAS and CMS and introduce a new photon
isolation criterion. This greatly increases the acceptance for light particles
at the expense of a somewhat larger Standard Model background. We show how
such an analysis would likely yield new experimental coverage for axion-like
particles for masses below 15 GeV, bridging the gap with the region covered by
flavor factories.

††preprint: CERN-TH-2021-216

###### Contents

  1. I Introduction
  2. II Analysis
    1. II.1 Photon identification and isolation
    2. II.2 Analysis cuts
    3. II.3 Signal
    4. II.4 Background
    5. II.5 Results
  3. III Theory motivation
    1. III.1 Example UV completion
    2. III.2 More complete models
  4. IV Discussion
  5. A Simulation details
    1. A.1 Background simulation
    2. A.2 Mass resolution

##  I Introduction

Diphoton resonances have long been recognized as a fertile ground for
potential new discoveries, most notably by the discovery of the Standard Model
(SM) Higgs boson in the $\gamma\gamma$ channel CMS:2012qbp ; ATLAS:2012yve .
The latter was of course anticipated before the construction of the LHC, and
as a result both ATLAS and CMS have excellent photon identification and
reconstruction capabilities. On the theory front, it is well known that any
new scalar or pseudoscalar particle with a coupling charged fermions will
typically generate a coupling to two photons at the one loop level. As was the
case for the SM Higgs, this naively subleading channel can be a primary
discovery mode if the decay widths to the SM fermions are suppressed or
background limited. These considerations have resulted in a number of powerful
searches, so far with sensitivity in the mass range between 65 GeV and 3 TeV
ATLAS:2014jdv ; CMS:2018dqv ; CMS:2018cyk ; ATLAS:2021uiz . Extending the
reach to lower invariant masses is feasible in the near future with present
data and proposals have already been put forward both for ATLAS and CMS
Mariotti:2017vtv ; CidVidal:2018eel  as well as LHCb CidVidal:2018blh .111Both
ATLAS and CMS have carried out a search for lighter diphoton resonances in
ultraperipheral heavy ion collisions CMS:2018erd ; ATLAS:2020hii . These
searches are however only sensitive if the branching ratio of the new state to
the $\gamma\gamma$ channel is $\mathcal{O}(1)$ Knapen:2016moh . In view of
future LHC runs, it is important to investigate how much further new trigger
techniques could extend the exploration of the diphoton spectrum at low
invariant masses.

The diphoton invariant mass can be written as

$\displaystyle m_{\gamma\gamma}$
$\displaystyle=\sqrt{2p_{T_{1}}^{\gamma}p_{T_{2}}^{\gamma}(\cosh\Delta\eta_{\gamma\gamma}-\cos\Delta\phi_{\gamma\gamma})}$
$\displaystyle\simeq\sqrt{p_{T_{1}}^{\gamma}p_{T_{2}}^{\gamma}}\Delta
R_{\gamma\gamma}\ ,$ (1)

where $p_{T_{1}}^{\gamma}$ ($p_{T_{2}}^{\gamma}$) is the transverse momentum
of the leading (subleading) photon, and $\Delta
R_{\gamma\gamma}=\sqrt{\Delta\eta_{\gamma\gamma}^{2}+\Delta\phi_{\gamma\gamma}^{2}}$
is their angular separation in the limit where both pseudorapidity separation
$\Delta\eta_{\gamma\gamma}\ll 1$ and azimuthal separation
$\Delta\phi_{\gamma\gamma}\ll 1$. This simple formula is instrumental to
understand why searches for low invariant mass resonances face severe
challenges related to the trigger system. The primary problem is the huge rate
of low $p_{T}$ photons produced in parton showers and in meson decays,
$\pi^{0}$ decays in particular. In order to keep the rate manageable, minimal
$p_{T}$ thresholds and photon isolation criteria are needed. Given Eq. I,
these result in a lower bound on the attainable diphoton invariant mass in a
given search.

For example, in CMS:2018cyk  CMS relied on a trigger which required a
$p_{T_{1}}^{\gamma}>30$ GeV ($p_{T_{2}}^{\gamma}>18$ GeV) threshold on the
leading (subleading) photon, tight isolation and an invariant mass requirement
of $m_{\gamma\gamma}>55$ GeV. The latter limited the accessible mass range in
the analysis to $m_{\gamma\gamma}>70$ GeV, once the turn-on of the trigger was
taken into account. To address this issue, ATLAS has implemented a special
trigger stream geared towards low mass diphoton resonances ATLAS:2019dpa . At
the hardware/level 1 trigger (L1 trigger), it requires two photons with a
symmetric $p_{T_{1,2}}^{\gamma}>15$ GeV requirement and loose isolation. This
L1 trigger seeds a software/high-level trigger (HLT) with
$p_{T_{1,2}}^{\gamma}>22$ GeV. This trigger combination collected already
138.5 $\text{fb}^{-1}$ data during the past LHC run, and it currently
represents the best way of exploring the low invariant mass region with ATLAS.
CMS has a similar, but asymmetric trigger with $p_{T_{1}}^{\gamma}>25$ GeV
($p_{T_{2}}^{\gamma}>12$ GeV) L1 thresholds for the leading (subleading)
photon, with and without a loose isolation criterion CMS:2020cmk .

Here we explore an alternative, forward-looking strategy to probe low mass
resonances, which we benchmark against the existing ATLAS trigger
ATLAS:2019dpa . The minimal photon $p_{T}$ and the diphoton angular separation
are interconnected by simple kinematics if we look at diphoton resonances, as
in Eq. I. For this reason, once the trigger thresholds on the photons are
accounted for, at low masses the resonances will be sufficiently boosted such
that the two photons can spoil each others isolation criteria. Our proposal
tries to address this problem in two ways.

First, we assess whether a strategy of directly using photons reconstructed at
the trigger level (called Turbo Stream Aaij:2019uij , Data Scouting
CMS:2016ltu  or Trigger-Level Analysis ATLAS:2018qto ) could yield a further
improvement relative to the existing ATLAS trigger ATLAS:2019dpa . In such a
strategy, only a small fraction of the full event record is written to tape,
which allows for a much higher output rate. The ATLAS, CMS and LHCb
collaborations have successfully deployed this technique for low-mass
resonance searches, in particular in the context of dijet CMS:2016ltu ;
ATLAS:2018qto  and dimuon PhysRevLett.124.131802 ; CMS-PAS-EXO-20-014 ;
LHCb:2019vmc  resonances. For the diphoton case, applying this technique could
theoretically allow for lower $p_{T}$ thresholds and possibly less stringent
isolation requirements, as compared to the normal trigger stream.

In addition we investigate whether the tension between the $p_{T}$ thresholds
and isolation cut can be further alleviated by defining a dedicated isolation
variable for a diphoton pair: when summing the $p_{T}$ of the activity within
the isolation cone, we will explicitly exclude the hardest, other photon
candidate. We also attempt to assess how the background from jets faking
photons behaves when subjected to this different isolation requirement. We
however do not attempt to modify the shower shape variables and our study can
thus be seen as a step towards a broader class of “photon-jet” objects at
trigger level Ellis:2012zp ; Allanach:2017qbs ; Sheff:2020jyw ; Wang:2021uyb ;
Ren:2021prq , where the isolation requirements and identification criteria on
the shower shape are tailored to the features of the targeted signal.

The experimental implementation of an analysis like what we propose here is
both a very time- and resource-consuming endeavour, and we therefore seek to
first estimate its possible gain with a phenomenological study. Our results
will hopefully aid the experimental collaborations in determining whether a
diphoton analysis directly at the trigger level is feasible and desirable,
given the resources at their disposal.

To quantify the possible gain in sensitivity, we use a well motivated, KSVZ
style Kim:1979if ; Shifman:1979if  axion-like particle (ALP) as an example
benchmark model, defined by

$\mathcal{L}_{a}\supset-\frac{1}{2}m_{a}^{2}a^{2}-\frac{\alpha_{s}}{8\pi}\frac{a}{f_{a}}G\tilde{G}+\frac{E}{N}\frac{\alpha_{\text{em}}}{8\pi}\frac{a}{f_{a}}F\tilde{F}\,.$
(2)

Here $N$ and $E$ are the anomaly coefficients for the gluon and photon
couplings respectively and the $a\to\gamma\gamma$ branching ratio is specified
by the ratio of $E/N$. In KSVZ models which are compatible with grand
unification one typically finds $E/N\sim\mathcal{O}(1)$, resulting in
Br($a\to\gamma\gamma)\sim 10^{-3}$ and Br($a\to jj)\sim 1$. Despite this
seemingly small branching ratio, the $\gamma\gamma$ channel is still a
superior signature due to the huge dijet background. The gluon coupling on the
other hand can yield very large cross sections for $m_{a}$ below 100 GeV,
partially offsetting the small branching ratio to photons.

The remainder of our paper is organized as follows: in Sec. II we describe our
simulation framework and analysis strategy, followed by a brief summary of the
theory motivation for low mass diphoton resonances in Sec. III. We conclude in
Sec. IV. In Appendix A we provide further details on our simulation framework.

##  II Analysis

In this section we describe our simulation framework and analysis strategy,
with a special emphasis on how our isolation criteria are defined. We take the
ATLAS calorimeter as a primary example, but most aspects of our study should
carry over to CMS.

###  II.1 Photon identification and isolation

In ATLAS Run-2 data taking the photon candidate has to satisfy reconstruction,
identification and isolation requirements which are described in ATL-PHYS-
PUB-2011-007 ; ATLAS:2019qmc ; ATLAS:2009zsq . Here we briefly review the main
steps, with a focus on how these requirements can limit the signal efficiency
for low mass diphoton resonances. First of all, the photon selection at L1 is
based on a reduced ECAL granularity. At present, a trigger ECAL tower is taken
to be a rectangle of $\Delta\eta\times\Delta\phi=0.075\times 0.098$
ATLAS:2016ecu .222The actual granularity of a single ECAL cell is
$\Delta\eta\times\Delta\phi=0.025\times 0.0245$ in the middle layer and goes
down to $\Delta\eta=0.0031$ if we consider the strip towers in first layer of
the ATLAS ECAL. This more refined information is however only available at the
HLT.

Offline, a photon candidate is classified as a (loosely) reconstructed photon
if a number of criteria are satisfied: first, the energy deposited in a
$0.25\times 0.25$ window of the hadronic calorimeter (HCAL) behind the ECAL
energy cluster has to be very small compared to the energy deposited in the
ECAL. (The ratio between HCAL and ECAL energy deposits is called hadronic
leakage in the experimental literature.) Second, ATLAS imposes requirements on
the shape of the electromagnetic shower in the middle ECAL layer, such as an
upper bound on the lateral width of the shower. In addition they define the
variables

$R_{\eta}\equiv\frac{E_{3\times 7}}{E_{7\times 7}}\quad\mathrm{and}\quad
R_{\phi}\equiv\frac{E_{3\times 3}}{E_{3\times 7}}.$ (3)

Here, e.g., $E_{3\times 7}$ is the energy deposited in a $3\times 7$ block of
ECAL cells, centered around the barycenter of the photon candidate, where the
cell are organized in an $\eta\times\phi$ grid. Both $R_{\eta}$ and $R_{\phi}$
are bounded from above, to ensure the shower of the photon candidate is
sufficiently contained. The above variables are used for loose reconstruction
and their primary role is to reject photons from collimated $\pi^{0}$ decays
ATL-PHYS-PUB-2011-006 ; ATL-PHYS-PUB-2011-007 ; ATLAS:2018fzd .

Loose (tight) photon isolation furthermore imposes an upper bound on the
amount of extra energy deposit within a cone of $\Delta R<0.2$ ($\Delta
R<0.4$) around the photon candidate. In particular, the ATLAS loose photon
isolation variable is defined as

$r_{\gamma}\equiv\frac{1}{p_{T_{\gamma}}}\Bigg{(}\\!\\!\\!\sum_{\begin{subarray}{c}\Delta
R<0.2\\\ p_{T_{i}}>0.5\rm{GeV}\end{subarray}}\\!\\!\\!\\!\\!p_{T_{i}}\Bigg{)}\
,$ (4)

and similarly for tight isolation. A photon is considered isolated if
$r_{\gamma}<0.05$.333In other studies, such as Ref. Catani:2018krb , isolation
is defined as an upper bound on the total transverse energy inside the
isolation cone, $E_{T}^{\text{MAX}}=r_{\gamma}p_{T_{\gamma}}$. The sum runs
over all tracks and calorimeter deposits with $p_{T}>0.5$ GeV and a distance
$\Delta R<0.2$ from the photon candidate, excluding the candidate itself. This
isolation criterion is meant to veto photon candidates that originated from
inside a jet, but it can also reject genuine, light diphoton resonances when
combined with the trigger requirements on the photon $p_{T}$.

To address this problem, we apply a trigger-level photon isolation criterion
where the photons are not counted towards each other’s isolation variable, as
will be explained in Eq. 5. We do not modify the other reconstruction and
identification criteria. As such, we still require that the minimal separation
between a diphoton pair is such that two L1 trigger ECAL towers do not
overlap, which roughly corresponds to $\Delta R_{\gamma\gamma}\gtrsim 0.1$. Of
course, a more detailed study by the experimental collaborations is required
to properly assess the behavior of the reconstruction and identification
algorithms in combination with the modified isolation defined here. For
instance, the use of the ATLAS $R_{\eta}$ and $R_{\phi}$ variables as defined
now may not be compatible with our modified isolation.

For our simulations, we rely on the Delphes 3 framework deFavereau:2013fsa ,
which propagates the particles of the event through a simplified detector
volume and assigns a fraction of the particle’s energy to a particular
collection of calorimeter cells. The candidate photons after this process are
identified with those ECAL cells for which there is no significant energy
deposit in the corresponding tower in the hadronic calorimeter and for which
there is no track pointing towards the ECAL cell.

To determine which photon candidates qualify as reconstructed photons, Delphes
3 uses the same isolation criterion as in Eq. 4. Candidates which fail
isolation are included among the inputs for the jet clustering algorithm
Cacciari:2011ma . Since we wish to preserve photon candidates which have
another hard photon candidate within their isolation cone, we modify the
standard isolation procedure by explicitly subtracting the $p_{T}$ of the
hardest photon candidate in the isolation cone from the sum. Concretely, we
modified the Delphes 3 isolation module to define a new isolation variable
$r_{\gamma\gamma}$, which is more suitable for photons which are expected to
come in pairs

$r_{\gamma\gamma}\equiv\frac{1}{p_{T_{\gamma}}}\Bigg{(}\\!\\!\\!\sum_{\begin{subarray}{c}\Delta
R<0.2\\\
p_{T_{i}}>0.5\rm{GeV}\end{subarray}}\\!\\!\\!\\!\\!p_{T_{i}}-p_{T_{\gamma,1}}\Bigg{)},$
(5)

where $p_{T_{\gamma}}$ is the transverse momentum of the primary photon
candidate, and $p_{T_{\gamma,1}}$ is the transverse momentum of the leading
photon candidate in the isolation cone, excluding the primary photon candidate
itself. The sum is defined as in Eq. 4. A photon candidate is then classified
as an isolated photon if it satisfies $r_{\gamma\gamma}<0.05$. A similar
modification of the isolation variable was already used in an offline analysis
ATLAS:2015rsn .

The isolation criterion described in Eq. 4 relies on tracking information.
However, tracking is a computing-intensive step, and it may therefore not be
available at high enough rates in an analysis that includes photons
reconstructed directly at the trigger level, due to the limited resources
available in the trigger farm. In the context of diphoton resonances, tracking
primarily improves the efficacy of pile-up subtraction methods and isolation
cuts, leading to a lower number of fake photons.444Tracking is also important
to extract data driven estimates for the photon fake rate ATLAS:2011gau . If
tracking is not available for the main trigger stream, a separate prescaled
stream may be needed to obtain the required control samples. We therefore
expect larger backgrounds for an analysis which cannot rely on tracking. To
estimate how much a search would degrade in the absence of tracking when
computing the isolation, we also repeat the procedure outlined above while
setting the tracking efficiency for all charged particles to zero.

###  II.2 Analysis cuts

As the implementation of our signal and the background simulations are
informed by our fiducial cuts, we first discuss those here. We must make a
number of assumptions, in particular in relation to which $p_{T}$ cuts may be
viable for a trigger-level analysis. For guidance, we look at the ATLAS L1
thresholds for two electromagnetic objects ATLAS:2019dpa , which where set to
$p_{T}>10$ GeV and $p_{T}>15$ GeV in the 2015 and 2016 data sets respectively
(L1_2EM10VH and L1_2EM15VH). As of the 2017 data set, a loose isolation
requirement was added (L1_2EM15VHI). We will here optimistically consider
$p_{T}>15$ GeV but assume no isolation requirement at L1, effectively
mimicking the cuts of the L1_2EM15VH trigger item. We assume that the modified
isolation criteria described in the previous subsection would be implemented
online after the L1 trigger. The rate for the L1_2EM15VH selection was
measured to be roughly 5 kHz ATLAS:2019dpa , which is similar to what was
recorded for an existing trigger-level analysis CMS-PAS-EXO-20-014 .

We estimate the expected sensitivity for the following three cases:

  * (I)

Trigger-level with tracking information, modified isolation (Eq. 5), and
requiring $p_{T\gamma}>15$ GeV for each photon, denoted by ‘Tracking’.

  * (II)

Same as above but without tracking, denoted by ‘No tracking’.

  * (III)

As a stand-in for an offline analysis with the existing ATLAS trigger, we
require $p_{T\gamma}>25$ GeV for each photon ATLAS:2019dpa .555We choose
$p_{T\gamma}>25$ GeV, as compared to the $p_{T\gamma}>22$ GeV in the ATLAS HLT
trigger strategy ATLAS:2019dpa , to approximately take into account the turn-
on of the trigger. For this case we assume standard isolation (Eq. 4) with
tracking information. Going forward, we will refer to this as the ‘Offline’
case.

Each event is required to contain two isolated photons with pseudorapidity cut
$|\eta_{\gamma}|<2.5$, each above the $p_{T}$ threshold as mentioned above.

Figure 1: Fiducial cross sections for the $a+$jet (solid) and $a$ (dot-dashed)
inclusive cross sections, for the selections described in the text. We switch
between the two samples when $m_{a}$ is twice the photon $p_{T}$ threshold.
The discontinuity can be resolved as a smooth step transition by matching the
samples at next-to-leading order, which we did not attempt here. For
$m_{a}\lesssim 10$ GeV, the photons are too collimated to pass the isolation
criterion for the “offline” selection.

###  II.3 Signal

The ALP benchmark model was implemented in Feynrules Alloul:2013bka , to
facilitate the generation of Monte Carlo sample with Madgraph 5 Alwall:2011uj
at leading order. The showering and hadronization was done with Pythia 8
Sjostrand:2006za ; Sjostrand:2007gs . We assumed a KSVZ ALP with vector-like
matter in a set of $5$-$\bar{5}$ multiplets, compatible with $SU(5)$
unification. This choice corresponds to $E/N=8/3$ in relation to Eq. 2 and is
only relevant for the $a\to\gamma\gamma$ branching ratio. The
$a\to\gamma\gamma$ branching ratio is sensitive to NNLO corrections to gluon
width Chetyrkin:1998mw  and has therefore a mild dependence on $m_{a}$. For
$5\,\text{GeV}<m_{a}<100\,\text{GeV}$, it is well approximated by the
phenomenological formula

Br $\displaystyle(a\to\gamma\gamma)\approx 7.8\times
10^{-4}\times\left(\frac{E/N}{8/3}\right)^{2}$
$\displaystyle\times\left[1+0.67\log\left(\frac{m_{a}}{10\,\text{GeV}}\right)+0.12\log\left(\frac{m_{a}}{10\,\text{GeV}}\right)^{2}\right].$
(6)

The detector response was simulated with Delphes 3, with the standard ATLAS
card, which we modified to more accurately represent the angular resolution of
the ATLAS ECAL (see Appendix A.2). We assumed an average of 40 pile-up
collisions per event and employed basic pile-up mitigation strategies, as
built into Delphes 3. Concretely, all tracks with a longitudinal separation of
more than $0.1$ mm from the primary vertex are removed and a pile-up density
correction is applied to the photon isolation variable deFavereau:2013fsa .

At low $m_{a}$, the most relevant process is an ALP produced in gluon fusion
together with an associated jet. The latter is needed to ensure that the
photons from the ALP decay can pass the trigger $p_{T}$ thresholds. The $a$ \+
jet production process was modeled at the level of the hard matrix element,
using Madgraph 5 with a minimal cut on the jet $p_{T}$: $p_{T,j}>30$ GeV as in
Hook:2019qoh . The fiducial rate does not increase if a fully inclusive
(matched) sample is used instead, as long as $m_{a}$ is less than twice the
photon $p_{T}$ threshold. We also verified that matching up to two jets does
not significantly change the relevant differential distributions. Following
Gershtein:2020mwi , we then compute the $a+1j$ cross section at the NLO with
Madgraph@NLO Alwall:2014hca ; Artoisenet:2013puc . The fiducial cross sections
subject to the cuts specified in Sec. II.2 are shown in Fig. 1 (solid lines).

For $m_{a}$ larger than twice the photon $p_{T}$ threshold no recoiling jet is
needed to satisfy the trigger requirements. An inclusive signal sample without
demanding an additional jet is therefore more suitable, as indicated by the
dot-dashed lines in Fig. 1. The sample was rescaled to match the NNLO cross
section in Mariotti:2017vtv , which was computed with the ggHiggs v3.5 package
Ball:2013bra ; Bonvini:2014jma ; Bonvini:2016frm . A smooth interpolation
between both regimes can be achieved by matching the zero jet and one jet
sample at NLO, which we did not attempt here.

The main take-away from Fig. 1 is that the “offline” selection starts to loose
acceptance for $m_{a}\lesssim$ 15 GeV, whereas both the “tracking” and “no
tracking” selections maintain reasonable acceptance as low as $m_{a}\approx 5$
GeV. We also see from Fig. 1 that the signal efficiency does not degrade much
in the absence of tracking, as expected.

###  II.4 Background

Figure 2: Stacked invariant mass distribution for the signal and the
background for the three scenarios described in Sec. II.2. A representative
signal resonance is shown for $m_{a}=10$ GeV with total number of events
corresponding to $f_{a}=100$ GeV. For the same plot with a larger mass range,
see Fig. 7 in the appendix. The ‘$\gamma j$ background’ in the panel labeled
‘Offline’ uses larger bin widths due to more limited simulation statistics.
See Sec. II.4 for details.

One of main backgrounds are events with pairs of relatively collimated
photons, which recoil against a jet with moderate $p_{T}$. In the SM, photons
arise either from the hard scattering matrix element, emission of a fermion
line in the parton shower or in hadron decays, notably $\pi^{0}$ decays.
Isolation criteria are designed to single out the former contribution, which
give us the most insight in the short distance physics. The shower and
hadronization contributions are captured in the perturbative and non-
perturbative components of the photon fragmentation functions, e.g. see
Bourhis:1997yu , and are ideally counted towards their corresponding jet. This
is achieved to a very high degree with the use of suitable isolation criteria.
However, even very small fake probabilities (probability that an isolated
photon arises through showering or hadronization) can be important due to the
comparatively high dijet and photon plus jet cross sections. This means that
backgrounds from jets faking photons cannot a priori be neglected.

We opt to simulate all components with a combination of Madgraph 5 and Pythia
8, as described below. We validate our simulation against the NNLO calculation
by Catani et al. Catani:2018krb , as well as experimental data ATLAS:2011gau
(see also ATLAS:2021mbt  for a recent update at $\sqrt{s}=13\text{ TeV}$).
This is described in Appendix A.1.

Photons originating from the hard collision were modeled with an inclusive
$\gamma\gamma$ sample, matched up to two jets. The events were then passed
through Delphes 3 as described in the previous sections. The collinear photon-
quark singularity was regulated with an angular cut of $\Delta
R_{q\gamma}>0.05$ at the generator level. The overall cross section was
rescaled to the NLO inclusive $\gamma\gamma$ cross section as computed with
Madgraph@NLO. The resulting invariant mass distributions are shown in Fig. 2,
labeled as “$\gamma\gamma$ background”, for our three different benchmark
strategies, as described in Sec. II.2.

A second background component arises from photon plus jet production, where
the jet fakes an additional photon. The most dominant contribution is the
$qg\to q\gamma$ process, whenever an additional hard photon is emitted during
the fragmentation of the outgoing quark. Fragmentation photons may be
identified as isolated if they are radiated at a wide angle from their
corresponding quark line. Such fake photons are relatively rare, however it is
essential to estimate their rate, because the $\gamma$ plus jet cross section
is much larger than the $\gamma\gamma$ cross section. This is partially due to
the additional $\alpha_{\rm em}$ suppression of the latter, but also due to
luminosity functions: at leading order, the $\gamma\gamma$ process requires a
$q\bar{q}$ initial state, whose luminosity function is much smaller than the
$qg$ luminosity.

To characterize this contribution it is helpful to look at the kinematics of
the $q\to q\gamma$ splitting function: In the colinear limit the angle between
the photon and the outgoing quark is given by

$\theta_{\gamma q}\approx\frac{Q}{E_{q}}\sqrt{\frac{1}{z(1-z)}}\ ,$ (7)

where $Q$ and $E_{q}$ are the virtuality and energy of the incoming quark
respectively, while $z=E_{\gamma}/E_{q}$ is the energy fraction carried away
by the photon. The Pythia 8 parton shower is ordered according to virtuality,
which through Eq. 7 implies that wide angle radiation occurs primarily in the
initial stages of the parton shower. However, emissions with $\Delta
R_{q\gamma}>0.05$ populate the same region of phase space as the hard process
simulation described in the previous paragraph. We therefore veto them to
avoid double counting. The effect of this veto is shown in Fig. 3, where we
show the $\Delta R_{q\gamma}$ distribution for fragmentation photons,
subjected normal isolation, modified isolation and no isolation. Photons from
the non-perturbative parts of the fragmentation process, e.g., hadron decays,
are accounted for separately and not included in this plot. As expected, all
three distributions align for $\Delta R_{q\gamma}$ larger than the isolation
cone, while the two isolated distributions drop rapidly for $\Delta
R_{q\gamma}<0.4$. Since all photons with $\Delta R_{q\gamma}>0.05$ are veto-
ed, we find that this contribution to the isolated photon distribution can be
neglected.

Figure 3:  Angular separation of hard ($p_{T}>15$ GeV) fragmentation photons
from their sister quark in a leading order $gq\to q\gamma$ sample, simulated
with Pythia 8 and rescaled to match the NLO, inclusive photon plus jet cross
section. Only photons from the perturbative parton shower are included. The
veto cut and isolation cone are indicated for reference. (See text for
details.)

Fake photons can also be produced in the non-perturbative part of the quark
fragmentation function, primarily from $\pi^{0}$ decays. To estimate this
part, we simulate a $pp\to\gamma j(j)$ sample with Madgraph 5, matched up to
two jets. The cross section was rescaled to match the inclusive, NLO $\gamma
j$ cross section as computed with Madgraph@NLO. The events were subsequently
showered and hadronized with Pythia 8. In this step all electromagnetic
radiation within the jet was veto-ed, as we separately accounted for this
component in the previous paragraph. The resulting distribution is shown in
Fig. 2, labeled as “$\gamma j$ background”. It is comparable or larger than
the true photon background for our modified isolation criterion, and smaller
than the true photon background for standard isolation. The fake photon
background can be further reduced by imposing tighter isolation cuts; we
comment on this in Appendix A.1.

In our discussion above we have always assumed that at least one photon
originated from the hard matrix element. A priori it is possible that both the
photons are supplied by the fragmentation of quarks in multi-jet processes.
This was found to be negligible by Catani et al. Catani:2018krb , although in
a slightly different kinematical regime. We verified that this conclusion
extends to our kinematical cuts, as long as the fragmentation probability for
both photons are assumed to be uncorrelated. Whether this assumption holds for
the low $m_{\gamma\gamma}$ regime of interest must be verified with data.

Finally, in a realistic experimental environment fake photons also arise from
electrons for which the track was not reconstructed or from hadrons which stop
in the first part of the ECAL. Both are difficult to model with a theorist-
level simulation and they are neglected in this study.

###  II.5 Results

As expected, Fig. 2 makes it clear that both signal and background go up
substantially for a trigger-level analysis, as compared to the “offline”
scenario. A trigger-level approach is most clearly useful for $m_{a}\lesssim
10$ GeV, as here the signal efficiency of the “offline” selection dies off
rapidly, as seen from Fig. 1. For $m_{a}\gtrsim 10$ GeV the relative power of
both approaches will depend on a number of important experimental subtleties,
such as the precise thresholds and turn-on of the various triggers as well as
various sources of systematic uncertainties.

To produce an estimate for the reach of both strategies, a number of
additional assumptions are therefore needed. In particular, we will assume
that the dominant uncertainty on the background is statistical rather than
systematic. This is a reasonable assumption for a bump-hunt over a smooth
background, however, any features in the invariant mass distribution of the
diphoton background below the trigger threshold might introduce further
complications. We also do not attempt a full fledged likelihood analysis and
instead simply compare the number of signal and background events in bins with
width set to the expected invariant mass resolution (See Appendix A.2). This
can be justified because of the large values of expected signal and background
events. For example, as can be seen in Fig. 2, the number of background and
signal events (for $m_{a}<100$ GeV and $f_{a}<1$ TeV) in a single invariant
mass bin containing the signal is $\sim 10^{4}$ and $\sim 10^{2}$
respectively. One can first construct a test statistic Cowan:2010js

$\displaystyle t=-2\ln\left(\frac{L(n|b)}{L(n|s+b)}\right)$ (8)

to obtain the significance. Here $L(n|m)$ denotes the probability of observing
$n$ events originating from a Poisson distribution with mean $m$, and $s$ and
$b$ denote the expected number of signal and background events respectively.
By assuming $n=s+b\gg 1$ and expanding in $s\ll b$, $t$ can be written as

$\displaystyle
t=2\left(n\ln\left(\frac{s+b}{b}\right)-s\right)\approx\frac{s^{2}}{b}.$ (9)

In the limit of large number of samples, $t$ is $\chi^{2}$-distributed with
one degree of freedom under the null hypothesis. Therefore, for each value of
$m_{a}$, we consider a single invariant mass bin which contains $90\%$ of the
signal and determine $f_{a}$ such that $\sqrt{t}=2$, to obtain a $2\sigma$
confidence limit.

With this simplified recipe we find similar sensitivity for all approaches for
$m_{a}\gtrsim 10$ GeV but see that a trigger-level analysis can potentially
uniquely probe the $m_{a}\lesssim 10\text{ GeV}$ region, as shown in Fig. 4.
While the overall normalization of our estimate is sensitive to all the
subtleties listed above, we emphasize that this feature should be robust, as
it is driven by the behavior of the signal efficiency in Fig. 1. All
strategies would however be a significant improvement over the existing
bounds.

Finally, we comment on the possibility to observe the Standard Model
$\eta_{b}\to\gamma\gamma$ decay.666We thank Michelangelo Mangano for pointing
this out to us. The $\eta_{b}$ is the lightest state in the bottomonium system
with mass $m_{\eta_{b}}\approx 9.4$ GeV. For a slightly boosted $\eta_{b}$
with $p_{T}\approx 20$ GeV, its production cross section at the LHC is
expected to be between 1 and 10 nb Likhoded:2017kfw . The $\eta_{b}$ branching
ratio to photons has not been measured yet but is thought to be roughly
$5\times 10^{-5}$Fabiano:2003yk . This rate might be high enough to be
observable in a trigger-level analysis, depending on the efficiency of our
isolation requirements. We leave this interesting question for a future study.

Figure 4: Estimated 2$\sigma$ exclusion reach for 100 $\text{fb}^{-1}$, along
with existing constraints from ATLAS ATLAS:2014jdv ; Mariotti:2017vtv , CMS
CMS:2017dcz ; CMS-PAS-HIG-14-037 ; CMS-PAS-HIG-17-013 ; CMS:2017dcz ;
Mariotti:2017vtv , LHCb CidVidal:2018blh  and exotic $B$ to $a$ decays
Chakraborty:2021wda . The smooth green contour represents the bound on the
inclusive $B\to sa$, while the remaining green contours represent several
exclusive channels. These bound are somewhat model dependent, see
Chakraborty:2021wda  for details. We also show an aggressive projection for
LHCb, assuming 300 $\text{fb}^{-1}$ CidVidal:2018blh . The right-hand $y$-axis
indicates the mass of the colored fermions ($\psi$) for the example UV
completion described in Sec. III.1.

##  III Theory motivation

We first comment on the generic features of the UV completions of the
effective theory in Eq. 2. Then we briefly review some of the top-down
motivation for low mass diphoton resonances.

###  III.1 Example UV completion

For completeness, we should investigate whether there exist healthy UV
completions of the effective action in Eq. 2, given the values of $f_{a}$ that
are probed in Fig. 4 and the complementary searches at the LHC probing new
physics at this scale.

As a minimal example we take the model by Kim, Shifman, Vainshtein and
Zakharov (KSVZ) Kim:1979if ; Shifman:1979if , where the ALP is the phase of a
complex scalar singlet $\Phi$. The model also includes a set of heavy, vector-
like colored and electrically charged fermions that are charged under the PQ
symmetry. This specifies the action

$\mathcal{L}_{\text{KSVZ}}\supset
g_{*}\Phi\tilde{\psi}\psi,\quad\quad\Phi=\frac{v_{a}+\varphi}{\sqrt{2}}e^{ia/v_{a}}.$
(10)

In our conventions, the vacuum expectation value of $\Phi$ ($v_{a}$) is
related to the ALP decay constant by $v_{a}=2Nf_{a}$. The anomaly coefficients
$N$ and $E$ are specified by the multiplicity and the representation of the
fermions. Taking the $\psi$ to fill out a $5$-$\bar{5}$ representation of
$SU(5)$, we have $N=N_{\psi}/2$ and $E=4/3N_{\psi}$ with $N_{\psi}$ the number
of $\psi$ flavors. Finally, the $g_{*}$ coupling determines the mass of the
fermions $m_{\psi}=N_{\psi}g_{*}f_{a}/\sqrt{2}$. Taking for example
$N_{\psi}=5$ and $g_{*}=3$ ensures that the $\psi$ states can be well heavier
than the reach of the many LHC searches for new colored particles. (See right
hand $y$-axis in Fig. 4). This example indicates that the discovery of a low
mass diphoton resonance would likely be the hallmark of an elaborate composite
sector near the TeV scale.

Alternatively, it is not difficult to increase the branching ratio to photons
in Eq. II.3 by choosing exotic representations which yield a larger $E/N$
ratio. One can also consider low values of $f_{a}$ but lift the masses of the
colored particles in the multiplet by means of an additional mass term in Eq.
10. This does not reduce the overall signal rate, since the reduced production
cross section is exactly compensated by an increase in the branching ratio to
photons, as long as the decay width to photons remains small compared to the
decay width to gluons. This remains true as long as the mass of the colored
particles satisfies $M\lesssim(\alpha_{s}/\alpha_{\rm em})\,g_{\ast}f_{a}$.
For the parameters in Fig. 4, this easily pushes the colored states outside
the energy reach of the LHC.

###  III.2 More complete models

The ALP benchmark model defined in Eq. 2 can be the hallmark of more complete
models addressing the hierarchy, the strong CP or the dark matter problems.
These three motivations were reviewed comprehensively in CidVidal:2018eel ;
Gershtein:2020mwi , and we therefore only summarize the main points.

The main virtue of very low scale supersymmetry breaking is that a viable
gravitino abundance can be achieved without the need for a non-standard
cosmological history Osato:2016ixc . In such scenarios, one also expects its
associated $U(1)_{R}$ symmetry to be broken at a low scale, leading to a
parametrically light pseudo-goldstone boson, the $R$-axion Nelson:1993nf ;
Intriligator:2007py ; Bellazzini:2017neg . The $R$-axion couples to the
gauginos, which induce the operators in Eq. 2 in the low energy theory.

Axions famously provide a solution to the strong CP problem by promoting the
QCD $\theta$-angle to a dynamical field Peccei:1977hh ; Peccei:1977ur . This
mechanism however comes with a _PQ quality problem_ , which states that any
sources of PQ breaking other than the QCD dynamics must be very small. The
suppression of potentially dangerous Planck-suppressed operators can achieved
through extending the UV structure of the model Izawa:2002qk ; Fukunaga:2003sz
; Cheng:2001ys ; Redi:2016esr  or by choosing the axion decay constant
sufficiently low. The latter option is already excluded in its most minimal
form Raffelt:1985nk ; Raffelt:1987yu ; Lai:2002kf ; Artamonov:2005ru ;
Abouzaid:2008xm ; Ceccucci:2014oza , though the constraints can be evaded in
models which raise the axion mass without affecting the alignment of its
potential with the CP preserving vacuum Rubakov:1997vp ; Berezhiani:2000gh ;
Hook:2014cda ; Fukuda:2015ana ; Dimopoulos:2016lvn ; Holdom:1982ex ;
Choi:1988sy ; Holdom:1985vx ; Dine:1986bg ; Flynn:1987rs ; Choi:1998ep ;
Agrawal:2017ksf ; Agrawal:2017evu ; Hook:2019qoh . The collider phenomenology
of such models maps directly onto the simplified model in Eq. 2 if the axion
is of the KSVZ class Kim:1979if ; Shifman:1979if . In the case of Dine-
Fischler-Srednicki-Zhitnitsky axions Dine:1981rt ; Zhitnitsky:1980tq  coupling
to SM leptons are also present, even though the best sensitivity at these
masses might still come from the diphoton final state CMS:2012fgd ;
CMS:2019buh ; LHCb:2018cjc ; Cacciapaglia:2017iws ; BuarqueFranzosi:2021kky .

Finally, the ALP in Eq. 2 could couple to the dark matter particle and be
responsible for its freeze-out into SM states. A natural realization of this
model with $\mathcal{O}(1)$ coupling constants predicts $f_{a}\sim$TeV
CidVidal:2018eel .

In the above, we have always assumed that the diphoton resonance is directly
produced in the partonic collision, typically associated with a gluon. This
need not to be the case however, and such a new state could also be produced
in the decay of a heavier state, such as the Higgs. Specifically, if we
consider a scalar field $S$ with couplings

$\mathcal{L}\supset\frac{\lambda_{S}}{2}S^{2}H^{\dagger}H+\frac{\alpha_{\rm
em}}{4\pi}\frac{1}{\Lambda_{\gamma\gamma}}SF^{\mu\nu}F_{\mu\nu}+\frac{\alpha_{s}}{4\pi}\frac{1}{\Lambda_{gg}}SG^{\mu\nu}G_{\mu\nu}\
,$ (11)

where we can neglect possible couplings to the $WW$, $ZZ$ and $Z\gamma$
operators as long as $m_{S}\ll m_{W}$. This simplified model reproduces exotic
Higgs decays of the form $h\to 4\gamma$, $h\to 2\gamma 2g$ and $h\to 4g$,
whose relative importance depends on $\Lambda_{\gamma\gamma}/\Lambda_{gg}$.
The four gluon channel is likely hopeless, but ATLAS and CMS have already
performed searches for the $h\to 2\gamma 2g$ ATLAS:2018jnf  and $h\to 4\gamma$
ATLAS:2015rsn ; CMS-PAS-HIG-21-003  channels. These searches place the
following constraints for $m_{S}\approx 20$ GeV

$\displaystyle\mathrm{Br}(h\to
SS)\times\mathrm{Br}(S\to\gamma\gamma)\times\mathrm{Br}(S\to gg)\lesssim 0.1$
$\displaystyle\mathrm{Br}(h\to
SS)\times\mathrm{Br}(S\to\gamma\gamma)^{2}\lesssim 2\times 10^{-4}.$ (12)

In the context of the simplified model in Eq. 11, the $h\to 4\gamma$ channel
is therefore always more powerful and no trigger-level analysis in the
diphoton topology is needed.

The reason is the $\mathrm{Br}(S\to gg)+\mathrm{Br}(S\to\gamma\gamma)\leq 1$
constraint, which is however easily evaded by adding an additional scalar

$\displaystyle\mathcal{L}\supset$
$\displaystyle\lambda_{S}S_{1}S_{2}H^{\dagger}H+\frac{\alpha}{4\pi}\frac{1}{\Lambda_{\gamma\gamma}}S_{1}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle+\frac{\alpha_{s}}{4\pi}\frac{1}{\Lambda_{gg}}S_{2}\mathrm{Tr}\left[G^{\mu\nu}G_{\mu\nu}\right].$
(13)

This model only induces the $h\to S_{1}S_{2}\to gg\gamma\gamma$ topology.
While we are not aware of a direct, top-down theory motivation for this
specific topology, Eq. III.2 is a perfectly plausible and fairly economical
extension of the SM Higgs sector. As such, it is important that it be covered
as well as possible, since Nature may very well be more clever (or devious)
than modern day model builders. The ATLAS search ATLAS:2018jnf  in particular
makes use of the diphoton trigger path discussed in the introduction, which
limits their reach to $m_{\gamma\gamma}\geq 20$ GeV. If a trigger-level
analysis for lower invariant masses proves to be feasible, the $h\to 2\gamma
2g$ topology should therefore benefit as well.

In addition to the above simple, two step cascade models more complicated
dynamics can be realized in hidden valley models Strassler:2006im ; Han:2007ae
, which can yield one or more low mass diphoton resonances Ellis:2012zp ;
Knapen:2021eip .

##  IV Discussion

Low mass diphoton resonances could hide in plain sight, being rejected by the
experimental trigger selections. We showed that the uncharted mass window
below the current reach of diphoton searches and above the existing
constraints from flavor physics can be probed by ATLAS and CMS. While probing
masses down to roughly 10 GeV is probably feasible with the current triggers,
we find that a trigger-level analysis with modified isolation requirements may
cover the whole mass region down to masses of order 1 GeV.

We considered a simple, modified isolation requirement which was already used
by ATLAS in a prior analysis ATLAS:2015rsn  and argue that it could
significantly increase the experimental acceptance for light resonances if
implemented in a trigger-level analysis. Though the increased signal
acceptance comes at the cost of a larger background rate, we nevertheless find
otherwise inaccessible parameter space. That said, there are likely major
technical challenges associated with a high rate, trigger-level analysis which
can only be assessed to a satisfactory level by the collaborations themselves.
The size of the fake photon background in particular should be verified in
data before a definitive claim can be made.

We remark that our analysis strategy is deliberately simplistic, and perhaps
even naive, given the expected computational constraints on a trigger-level
analysis. A number of more sophisticated analysis strategies have already been
proposed for scenarios where the trigger is not the main challenge, most
notably the topology $h\to 2a\to 4\gamma$ Dobrescu:2000jt ; Draper:2012xt  but
also cascade decays from heavier, exotic resonances Toro:2012sv . Jet
substructure and machine learning techniques applied to the energy depositions
in the photon isolation cone appear to be promising Ellis:2012zp ;
Allanach:2017qbs ; Sheff:2020jyw ; Wang:2021uyb ; Ren:2021prq . Should it be
possible to implement some of these techniques in a trigger-level analysis,
they should yield substantially better sensitivity than our comparatively
simplistic isolation requirement.

A more ambitious possibility would be to push to even lower
$m_{\gamma\gamma}$, to the extent that both photons merge and one must rely on
the single photon L1 seed. This effectively involves modifying the photon
identification requirements themselves. This was already accomplished in an
(offline) ATLAS ATLAS:2018dfo  search for a pair of photon jets. However the
pay-off could be very substantial if this can also be done for a single photon
jet as a trigger-level analysis, given that the single photon trigger jumps
from 22 GeV to 120 GeV between L1 and HLT ATLAS:2019dpa . While the technical
challenges should not be underestimated, we conclude that trigger-level
searches for low mass diphoton resonances and photon jets could open
qualitatively new parameter space for the discovery of low mass diphoton
resonances.

###### Acknowledgements.

We thank Liron Barak, Matt Low, Alberto Mariotti, Marco Montella, José Ocariz,
Michele Papucci, Luis Pascual, Dean Robinson, Filippo Sala and Kohsaku Tobioka
for many discussions, comments on the draft and earlier work on similar
topics. We thank Christian Bauer, Zoltan Ligeti, Michelangelo Mangano and
Nicholas Rodd in particular for discussions on the details of the photon
fragmentation functions. We are especially grateful to Caterina Doglioni for
the discussions that initiated this project and for detailed comments on the
manuscript. SKn was supported by the Office of High Energy Physics of the U.S.
Department of Energy under contract DE-AC02-05CH11231. SKu was supported in
part by the National Science Foundation (NSF) grant PHY-1915314 and the U.S.
Department of Energy Contract DE-AC02-05CH11231. SKu thanks Aspen Center of
Physics, supported by NSF grant PHY-1607611, for its hospitality while this
work was in progress. All authors thank the Galileo Galilei Institute for
Theoretical Physics for its hospitality and the INFN for partial support
during the initial and final phases of this work.

##  Appendix A Simulation details

###  A.1 Background simulation

To validate our simulation framework, we compare with the NNLO calculations
performed by Catani et al. Catani:2011qz ; Catani:2018krb , which have shown
good agreement with ATLAS data at 7 TeV ATLAS:2012fgo . This comparison is
shown in Fig. 5. Note that these results are obtained with a fixed isolation
threshold $E_{T}^{\rm MAX}=2$ GeV, rather than with the sliding threshold we
used in Eq. 4 and Eq. 5. The leading and subleading photon must satisfy
$p_{T}>25$ GeV and $p_{T}>22$ GeV respectively, as well as a rapidity cut
$|y_{\gamma}|<2.37$. At leading order, the process is just the
$q\bar{q}\to\gamma\gamma$, for which there is no phase space for
$m_{\gamma\gamma}<50$ GeV. At NLO, new processes contribute such as
$qg\to\gamma\gamma q$ and $qg\to\gamma q$, where a second photon is supplied
by the quark fragmentation function. This contribution is enhanced by the
parton distribution functions (pdf’s), despite being formally higher order
than the $q\bar{q}\to\gamma\gamma$ process. The NLO calculation also contains
virtual corrections to $q\bar{q}\to\gamma\gamma$. The additional jet at NLO
opens up the phase space for $m_{\gamma\gamma}<50$ GeV. Finally, the NNLO
calculation allows for an additional soft jet and includes virtual corrections
via the box diagrams. The latter are important because they include the
$gg\to\gamma\gamma$ process, which is further pdf enhanced compared to the LO
and NLO contributions. The blue curve, diphoton events matched up to one jet,
in Fig. 5 is obtained with Madgraph@NLO and is in good agreement with the
Catani et al. NLO calculation.

For computational reasons, we used a leading order matched sample, rescaled to
the inclusive NLO cross section, as described in Sec. II.4. To estimate the
error this simplification introduces, we apply it to the kinematical
selections used by Catani et al., as is shown by the orange curve in Fig. 5.
Here we neglected photons from the non-perturbative part of the fragmentation,
which is a subleading component for standard isolation (see the rightmost
panel of Fig. 2). Our simplified procedure somewhat underestimates the peak
but is in $\mathcal{O}(1)$ agreement with the low $m_{\gamma\gamma}$ tail.
While this may seem far from ideal, it worth noting that the sensitivity
projection in Fig. 4 only scale as the $4^{\text{th}}$ root of the background.
Our qualitative conclusions are therefore not very sensitive to this level of
background mismodeling. For completeness we extend the invariant mass
distributions in Fig. 2 to the full mass range in Fig. 7. We clearly see the
feature around twice the $p_{T}$ threshold, consistent with the discussion in
the previous paragraph.

Figure 5: Diphoton invariant mass plot from Catani et al. Catani:2018krb ,
overlaid with our simplified procedure (orange, rescaled by an NLO K-factor of
2.8) and Madgraph@NLO (blue). See text for details.

Finally, we also comment on the dependence of the hadronic contribution to the
photon background on the choice of isolation criterion. For this purpose, we
simulated a large $q\bar{q}$ sample with Pythia 8 and quantify the probability
of a jet producing an isolated photon from a hadronic decay. To cleanly
separate this contribution from the perturbative part of the fragmentation
function we veto all electromagnetic emissions during the parton shower.
(These contributions were discussed separately in Sec. II.4, specifically in
Fig. 3.)

We find that with the modified isolation criterion, the fake rate increases by
a bit more than an order of magnitude. Using tight isolation does reduce the
background somewhat, but does not fully compensate for the aforementioned
increase. That said, given the rather special corner of phase space that is
being probed here, a data-driven approach will be essential to validate the
Monte Carlo predictions.

Figure 6: Probability of a quark jet fragmenting into an isolated photon with
$p_{T}>15$ GeV, for standard and modified isolation, with loose and tight
selections (see Sec. II.1). Only photons from hadronic decays are considered.
For photons from the perturbative part of the parton shower we refer to Fig.
3.

Figure 7: Stacked invariant mass distribution for the signal and the
background for the three scenarios described in Sec. II.2, for a larger mass
range. A representative signal resonance is shown for $m_{a}=10$ GeV with
total number of events corresponding to $f_{a}=100$ GeV.

###  A.2 Mass resolution

Given the limited number of handles at our disposal, the main background
discrimination comes from hunting for a bump on top of a large, background.
The resolution on the invariant mass is therefore very important. This is
handled automatically by Delphes 3, as shown in Fig. 8. Given its importance,
we seek to verify the built in Delphes 3 parametrization explicitly.
Concretely, the energy resolution of the ATLAS ECAL is parametrized by
ATLASElectromagneticBarrelCalorimeter:2006ymk

$\frac{\delta
E_{\gamma}}{E_{\gamma}}=10\%\left(\frac{\text{GeV}}{E_{\gamma}}\right)^{1/2}\oplus
0.7\%.$ (14)

where the $\oplus$ notation indicates that the errors are added in quadrature.
The mass resolution is given by

$\frac{\delta
m_{\gamma\gamma}}{m_{\gamma\gamma}}=\frac{1}{2}\left(\frac{\delta
E_{\gamma,1}}{E_{\gamma,1}}\oplus\frac{\delta
E_{\gamma,2}}{E_{\gamma,2}}\right)\oplus\frac{0.0247}{\Delta
R_{\gamma\gamma}}$ (15)

where for the angular term we have taken the off-line granularity of the
calorimeter. This is slightly worse than the true off-line resolution of the
ECAL but better than the L1 angular resolution. The assumption here is that
the fine grained information for the ECAL tower can be written out for a small
region of interest around the candidate resonance. To compare with Fig. 8, we
estimate $\delta m_{\gamma\gamma}$ by taking $\eta\approx 0$ for both photons
and saturating $E_{\gamma,1}$ and $E_{\gamma,2}$ to the $p_{T}$ thresholds.
This determines the energy resolution via Eq. 14. We then solve for $\Delta
R_{\gamma\gamma}$ in Eq. I by using the injected invariant mass. The resulting
numbers are plugged into Eq. 15 to obtain the estimated mass resolution. We
overlaid a Lorentzian line shape with width $\delta m_{\gamma\gamma}$ to
visualize the agreement with the full Delphes 3 simulation.

Figure 8: Mass resolution for a diphoton resonance, as reproduced by Delphes 3
with the standard ATLAS configuration card (solid). To determine the width of
the overlaid Lorentzian line shape we take a $p_{T}$ threshold of 15 GeV
(dashed), as described in the text.

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