MI-HET-805

# A Novel Beam-Dump Measurement with the LHC General-Purpose Detectors

Bhaskar Dutta<EMAIL_ADDRESS>Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy, Texas A&M University, College
Station, TX 77843, USA  Doojin Kim<EMAIL_ADDRESS>Mitchell Institute for
Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas
A&M University, College Station, TX 77843, USA  Hyunyong Kim
<EMAIL_ADDRESS>Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA

###### Abstract

We propose a novel scheme for performing a beam-dump-like experiment with the
general-purpose detectors (ATLAS and CMS) at the LHC. Collisions of high-
energy protons result in jets containing a number of energetic hadrons and
electromagnetic objects that are essentially ``dumped'' to hadronic and
electromagnetic calorimeters, respectively, and induce the production of
secondary hadrons, electrons, and photons in calorimetric showers. We envision
a situation where new physics particles are produced by the interactions of
these secondary particles inside the calorimeters. For proof of principles, we
consider the axion-like particles (ALPs) produced via the Primakoff process in
the presence of their interaction with photons at CMS. We argue that the drift
tube chambers and the ME0 module of the muon system can serve as detectors to
record the photons from the ALP decay, demonstrating that the resulting
sensitivity reach is competitive due to their close proximity to the signal
source points. We further show that the LHC does not suffer from a barrier,
dubbed beam-dump ``ceiling'', that typical beam-dump experiments hardly
surpass, carrying the great potential for exploring a wide range of parameter
space in increasing statistics. This analysis can be extended to investigate
various types of light mediators with couplings to the Standard Model leptons
and quarks.

Introduction. While the existence of dark matter in the universe is clearly
indicative of new physics beyond the Standard Model (SM), the experimental
effort in the search for dark-matter candidates, especially, GeV-scale weakly
interacting massive particles, via their hypothetical non-gravitational
interactions is not yet fruitful. As an alternative possibility, MeV-scale
dark-matter candidates are receiving great attention, as they can be thermally
produced as well and are less constrained by the existing searches. For them
to reproduce the observed dark-matter relic abundance, their interaction
strengths to the SM are likely to be feeble and MeV-scale mediators often come
into play. Therefore, the dark-matter search program is now promoted to a more
generic dark-sector search program including the mediators and their feebly
interacting nature motivates the intensity-frontier facilities such as beam-
based neutrino experiments.

By contrast, the general-purpose detectors (ATLAS and CMS) of the LHC, an
energy-frontier facility, has been designed to be optimally sensitive to GeV-
to-TeV-scale physics. The LHC is (indirectly) capable of probing MeV-scale
physics by placing additional forward-physics facilities distant from the
primary interaction point (e.g., FASER [1] and SND [2]). On the contrary, the
exploration of MeV-scale physics in the central region is still seemingly
unpromising and irrelevant.

We point out that the hadronic calorimeters (HCAL) and electromagnetic
calorimeters (ECAL) of the detectors can be considered as dumps of particles
inside a jet produced by a proton collision, sourcing MeV-scale new physics
particles as in typical beam-dump experiments. We then infer the production of
a new physics particle when it decays to SM particles inside the muon system.
The large production cross-section of jets ensures the copious production of
feebly-interacting new physics particles and the large angular coverage (i.e.,
almost $4\pi$ coverage in solid angle) of the muon system allows for a large
fiducial signal flux. Furthermore, high-energy beam protons allow new physics
particles to be energetic, hence significantly boosted, and the muon system is
extremely close to the calorimeters. Therefore, it is possible to access the
region of parameter space toward larger couplings and larger masses
(henceforth denoted by prompt-decay regime) that typical beam-dump experiments
would hardly reach.

We illustrate the main idea with an axion-like particle (ALP) interacting with
the SM photon:

$-\mathcal{L}_{\rm
int}\supset\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}\,,$ (1)

where $a$ and $F_{\mu\nu}$ ($\tilde{F}_{\mu\nu}$) denote the ALP field and the
(dual) field strength tensor of the SM photon and where $g_{a\gamma\gamma}$
parameterizes the interaction strength between the ALP and the SM photon. We
further assume that an ALP decays to a pair of photons that leave experimental
signatures at the CMS muon system.

Proposal outline. The proposed measurement begins with the production of jets
at the interaction point. A number of hadrons and electromagnetic particles
inside a jet are essentially ``dumped'' and absorbed to HCAL and ECAL,
respectively. They eventually end up with electromagnetic showers creating
photons (plus electrons and positrons) copiously, while potentially inducing
the production of secondary hadrons in the midst of absorption. These photons
can undergo the Primakoff process and convert to ALPs in the presence of a
non-zero $g_{a\gamma\gamma}$. The produced ALPs then enter a drift tube (DT)
chamber [3, 4] in the muon system and decay to photons therein. We also
consider ME0, a projected new station in the muon system available at Run4
[4], as it may readily allow for dedicated triggers for signal detection. We
will elaborate on the production of ALP events, the detection principle at DT
chambers and ME0, potential backgrounds, our sensitivity study, and the
trigger aspects in the next sections.

Signal production. A precise estimate of the photon flux inside HCAL and ECAL
is a key factor for a precise estimate of the sensitivity reach. In our study,
we first utilize Pythia8.2 [5] to simulate jet production. Long-lived hadrons
(e.g., $\pi^{\pm}$ and $K^{\pm}$) and stable particles (e.g., $e^{\pm}$ and
$\gamma$) are fed, respectively, into the simplified HCAL and ECAL modules
built under the GEANT4 [6] code package with the QGSP_BERT physics list which
simulates their subsequent interactions including electromagnetic showering.
We take simplified specifications that the HCAL and ECAL modules are mostly
Cu-based and PbWO4-based, respectively, and their effective average lengths
are set to be 1.3 m and 0.6 m, respectively.111By contrast, their widths and
heights are set to be large enough as their detailed scale does not affect the
photon flux substantially.

Denoting the photon energy by $E_{\gamma}$ and the angle of a photon with
respect to the beam axis by $\theta_{\gamma}$, we show the
$E_{\gamma}-\theta_{\gamma}$ correlations of the photons inside ECAL (left)
and HCAL (right) that are predicted by our GEANT4 simulations in FIG. 1. We
choose the photons satisfying $E_{\gamma}>1$ MeV and $\Delta
R_{\gamma}=\sqrt{(\Delta\theta_{\gamma j})^{2}+(\Delta\phi_{\gamma
j})^{2}}<0.5$ with $\Delta\theta_{\gamma j}$ and $\Delta\phi_{\gamma j}$ being
the polar and azimuthal angle distances between a given photon and the center
of the associated jet. By restricting photons to within the jet radius, we
isolate signals from potential backgrounds arising randomly outside the jet
cone. The number of photons is normalized to an integrated luminosity of 500
fb-1 at $\sqrt{s}=13.6$ TeV and the total numbers of photons are $1.1\times
10^{19}$ and $9.3\times 10^{18}$ for ECAL and HCAL, respectively. While the
majority of photons are soft ($\lesssim 10$ MeV) and forward-moving, a sizable
fraction of photons can be found in the central region. Also, we observe that
the forward-moving photons have a tendency to be harder than the central-
region photons.

Figure 1: Energy-angle ($E_{\gamma}-\theta_{\gamma}$) correlations of the
photons inside ECAL (left) and HCAL (right) satisfying $E_{\gamma}>1$ MeV and
$\Delta R_{\gamma}<0.5$. The number of photons is normalized to an integrated
luminosity of 500 fb-1 at $\sqrt{s}=13.6$ TeV. See the text for more details.

As mentioned earlier, a produced photon can convert to an ALP via the
Primakoff process with an atomic target (denoted by $N$) inside the
calorimeters, i.e., $\gamma+N\to a+N$, and the differential production cross-
section $\sigma_{P}$ in $\theta_{a}$, angle of the outgoing ALP with respect
to the incoming photon is given by

$\frac{d\sigma_{P}}{d\theta_{a}}=\frac{1}{4}g_{a\gamma\gamma}^{2}\alpha_{\rm
em}Z^{2}\left|F_{\rm
Helm}(t)\right|^{2}\frac{p_{a}^{4}\sin^{3}\theta_{a}}{t^{2}}\,$ (2)

where $\alpha_{\rm em}$, $Z$, and $p_{a}$ are the electromagnetic fine
structure constant, the atomic number of the target material, and the momentum
of the outgoing ALP, and momentum transfer
$t=m_{a}^{2}-2E_{\gamma}(E_{a}-p_{a}\cos\theta_{a})$. In the collinear limit
where the momentum transfer to the atomic system is negligible, $E_{a}\approx
E_{\gamma}$ and $\theta_{a}\to 0$. We take this limit in our data analysis,
i.e., the produced ALPs inherit the energy and momentum direction from the
incoming photons. Finally, $F_{\rm Helm}$ describes the usual Helm form
factor,

$F_{\rm
Helm}(t)=\frac{3j_{1}(\sqrt{|t|}R_{1})}{\sqrt{|t|}R_{1}}\exp\left(-\frac{|t|s^{2}}{2}\right),$
(3)

where $j_{1}$ is the spherical Bessel function, $s=0.9$ fm, and
$R_{1}=\sqrt{(1.23A^{1/3}-0.6)^{2}+2.18}$ fm with $A$ being the atomic mass
number of the target material [8].

Basically, a photon may go through either the Primakoff process or the other
SM processes (e.g., pair production and photoelectric absorption). Therefore,
the ALP production probability $P_{\rm prod}$ is given by

$P_{\rm prod}=\frac{\sigma_{P}}{\sigma_{\rm
SM}+\sigma_{P}}\approx\frac{\sigma_{P}}{\sigma_{\rm SM}},$ (4)

where we use the total cross-section of SM interactions $\sigma_{\rm SM}$
reported in Ref. [9] and the approximation holds for $\sigma_{\rm
SM}\gg\sigma_{P}$.

Detection principle. Once an ALP is produced, it should first travel to a
detector module of interest and decay therein. The detection probability, say
$P_{\rm det}$, is essentially governed by the exponential decay law and we
have

$P_{\rm
det}=\exp\left(-\frac{L}{\tilde{l}}\right)\left[1-\exp\left(-\frac{\Delta
L}{\tilde{l}}\right)\right],$ (5)

where $L$ is the distance between the ALP production point and the detector of
interest and $\Delta L$ is the travel length of the ALP within the detector of
interest. Here $\tilde{l}$ denotes the mean decay length that can be
calculated with the ALP boost factor $\gamma_{a}[=(1-\beta_{a}^{2})^{-1/2}]$
and the ALP decay width $\Gamma_{a}$:
$\tilde{l}=\beta_{a}c\cdot\gamma_{a}/\Gamma_{a}$ with $\Gamma_{a}$ given by

$\Gamma_{a}=\frac{1}{64\pi}g_{a\gamma\gamma}^{2}m_{a}^{3}\,.$ (6)

As mentioned earlier, we consider two types of detector modules in the muon
system for our sensitivity estimates:

  * •

DT chambers [3, 4]: DT chambers are located in $|\eta|<1.2$ of the barrel
region. Each of them consists of $8-12$ layers each of which is further
segmented into long aluminum drift cells filled with gas.

  * •

ME0 [4]: ME0 will be introduced to the muon system in the endcap region of CMS
to maintain its performance at the high-luminosity LHC. It has a 6-layer
structure and is based on the Gas Electron Multiplier technology and is
intended to cover the far-forward region of $2.0<|\eta|<2.8$.

Both modules are gas-based. Basically, gaseous detectors use the ionization,
drift, and diffusion processes to amplify signals generated by charged
particles passing through the gas volume.

The chamber system can detect photons, although it is primarily designed for
detecting muons. When an ALP enters a DT chamber, it can decay into two
photons inside a drift cell. Each of them then splits into an electron-
positron pair knocking electrons off the atoms of the gas. If such an electron
(or positron) is energetic enough, it can pass through several layers and fire
multiple drift cells, followed by getting absorbed into an iron yoke without
firing the drift cells inside the next DT chamber. Therefore, the expected
experimental signature is successively fired multiple drift cells belonging to
a series of layers, which collectively shows a diphoton pattern aligning with
the associated jet. We perform GEANT4 simulations, taking a simplified
geometry of multi-layered aluminum drift cells filled with argon gas and
injecting photons. The efficiency of photon detection depends on the energy of
the photons. Our simulation study suggests that photons with energies ranging
from a few MeV to GeV can pass through a single DT chamber and generate
multiple hits in the drift cells of consecutive layers.

The basic concept of signal detection at ME0 is essentially the same as that
of DT chambers. The photons from an ALP decay split into electron-positron
pairs which subsequently ionize the atoms of the gas and eventually result in
an electron avalanche recorded by some of the finely spaced readout strips.
Again if the initial electron (or positron) is energetic enough, it can pass
through several layers. Therefore, the expected experimental signature is a
set of fired readout strips in multiple layers, collectively forming a
diphoton pattern.

Background consideration. Given the detection principle, we identify two
possible background sources [10]: escaping hadrons and neutron gas. Their
collective rate is estimated to be $\sim 10^{-8}$/(dijet event) at DT chambers
[10]. Although HCAL is designed to capture and absorb hadrons, a small
fraction of them may escape from HCAL and continue to move into the muon
system. However, these hadrons are typically energetic enough to fire the
associated drift cells from the beginning of a given DT chamber to the end and
even multiple DT chambers. Likewise, they fire the associated readout strips
in every layer of ME0. The experimental signatures of this sort are clearly
distinguished from the signatures that typical ALP events would leave.
Therefore, given the small escape rate and the difference in the detector
signatures, we expect the escaping-hadron-induced backgrounds to be
negligible.

Neutron gas originates from radioactivated detector components and such
neutrons can induce spontaneous photons in a temporally and spatially random
manner that can further fire DT chambers or ME0. Since the typical energy of
these photons is within $0.1-10$ MeV [10], it is challenging for them to fire
more than a single drift cell in the detector modules of our interest. A
neutron-induced photon of a few MeV can fire a handful of drift cells, but it
is hard for it to accidentally create a diphoton pattern in association with
the jet of interest. Since expected signal events can fire multiple drift
cells and create an outward diphoton pattern, we expect that the neutron-gas-
induced backgrounds can be significantly suppressed to a negligible level.

Beam-dump ``ceiling'' and the LHC potential. Before presenting our main
results, we discuss the general features of sensitivity reaches in the beam-
dump-type experiments and the great potential of the LHC, especially focusing
on the prompt-decay regime. Typically in this regime, $L,\Delta L\gg\tilde{l}$
and Eq. (5) is approximated to $P_{\rm det}\approx\exp(-L/\tilde{l})$.222For
completeness purposes, we discuss the case of $L,\Delta L\ll\tilde{l}$ in
Supplement. Suppose that $N_{\gamma}$ photons set the sensitivity at
$g_{a\gamma\gamma}$ with $N_{\rm sig}$ ALP events.333For example, $N_{\rm
sig}=2.3$ is for the 90% C.L. sensitivity under a zero-background assumption.
If $x$-times larger data acquisition were expected, the new reach
$g_{a\gamma\gamma}^{\prime}$ under a zero-background assumption would be

$g_{a\gamma\gamma}^{\prime}\approx g_{a\gamma\gamma}\sqrt{1+\frac{\log
x}{\log(N_{\gamma}\langle P_{\rm prod}\rangle/N_{\rm sig})-1}}\,,$ (7)

where $\langle P_{\rm prod}\rangle$ denotes the average value of $P_{\rm
prod}$ at $g_{a\gamma\gamma}$; $N_{\gamma}\langle P_{\rm prod}\rangle\gg
N_{\rm sig}$ in typical situations. For a sufficiently large background, $\log
x$ is replaced with $\log x^{1/2}$. This implies that the sensitivity is
effectively no longer improved even with much larger statistics due to the
logarithmic behavior shown above.444This also implies that the signal
production rate per injection photon barely depends on $x$ as
$g_{a\gamma\gamma}$ defining the sensitivity does not change much. Due to this
observation, we ignored the dependence of the ALP production rate on
$g_{a\gamma\gamma}$ when deriving Eq. (7). Therefore, most of the beam-dump
experiments encounter this ``ceiling'' in probing the prompt-decay regime.
This argument is generically relevant to decay channels of long-lived
particles (e.g., dark photon, Higgs-portal scalar, etc).

However, this does not hold for the proposed beam-dump measurements at the
LHC. The reason is that as we will show shortly, for the
$(m_{a},g_{a\gamma\gamma})$ pairs defining the sensitivity lines in the
prompt-decay regime, the $L\gg\tilde{l}(\propto\gamma_{a})$ condition is
hardly satisfied; $L\sim\mathcal{O}(1-2\,{\rm m})$ and typical $\gamma_{a}$ is
very large under the LHC environment. Therefore, we expect that our proposal
does not suffer from the aforementioned beam-dump ``ceiling'' and possesses
great potential for exploring a wide range of the prompt-decay regime in
increasing statistics. This essentially motivates the development of dedicated
trigger algorithms to maximize signal statistics.

Results. We are now in the position to report our sensitivity estimates. Four
different scenarios are considered here: i) Run2 (150 fb-1) with the zero-bias
trigger [14], ii) Run2+3 ($150+250$ fb-1) with all high-level trigger [15],
iii) Run4 (500 fb-1) with a dedicated trigger, and iv) Run4 (500 fb-1) at ME0
with a dedicated trigger. Integrated luminosities of 250 fb-1 during Run3 and
500 fb-1 during Run4 are expected in 2025 and 2030, respectively [16]. In
i)-iii), DT chambers are the main detection modules while in iv) only ME0 is
considered. Regarding the trigger, in i), we conservatively consider events
collected by the existing zero-bias trigger with a prescale factor of
$1.1\times 10^{-6}$ [14]. In ii), we consider all high-level triggers most of
which involve energetic jets and/or electromagnetic objects. Our GEANT study
suggests that the spectral behavior of shower photons is similar modulo the
overall normalization, so we assume an effective zero-bias trigger with a
prescale factor enhanced by an order of magnitude that corresponds to the
total high-level trigger rate 1 kHz. Finally, in iii) and iv), we assume that
dedicated trigger algorithms are developed to maximize the acceptance of the
signal flux. For all scenarios, we further restrict ourselves to ALPs
converted from photons whose energy is greater than 5 MeV to ensure hits in
multiple layers of the detection modules (i.e., DT chambers and ME0).
Regarding the photon detection efficiency, based on the aforementioned GEANT
study, we assume idealized detection modules for simplicity, but our final
results are not appreciably affected by moderate degradation (i.e., a few
factors) of detection efficiency.

The 90% C.L. sensitivity reaches of the above-listed four scenarios are
displayed in FIG. 2 in the $(m_{a},g_{a\gamma\gamma})$ plane. Since we expect
negligible backgrounds, we require the number of signal events to be 2.3 on
the basis of the expected statistical error only. We also show the current
constraints from various existing experiments (e.g.,
$e^{+}+e^{-}\to\gamma+{\rm inv.}$ [18, 19, 20, 21, 22], Belle-II [23], CCM
[24], CHARM [25], E137 [26], E141 [27], LEP [28], NA64 [29], $\nu$Cal [30],
and PrimEx [31]), based on the limits compiled in e.g., Refs. [32, 33, 34].
The gray-colored region is based on the laboratory-produced ALP searches, and
the boundaries are set by various lepton collider and beam-dump experiments.
By contrast, the regions constrained by astrophysical considerations [35, 36,
37, 38, 39] (e.g., HB stars, supernova, etc.) are shown in yellow.555The
cosmological triangle region can be probed in various ongoing neutrino
experiments [43, 24]. The measurement of the explosion energy of SN1987A can
provide severe tension to this region unless the star cooling process is
significantly different from the standard picture [44].

Figure 2: Expected 90% C.L. sensitivity reaches of the ALP decay signal at the
detector modules of interest at CMS, under the assumption of negligible
backgrounds. Existing laboratory-based limits [18, 19, 20, 21, 22, 23, 25, 26,
27, 28, 29, 30, 31, 24] are shown in gray, while astrophysics-based limits
[35, 36, 37, 38, 39] are shown in yellow.

Our sensitivity study suggests that CMS can explore the regions beyond the
current beam-dump limits which none of the existing (in)direct experiments
have ever probed because ALPs belonging here would decay too promptly to reach
their detector system. Note that the detector modules of interest, i.e., DT
chambers and ME0, are within just a few meters from the signal production
points, i.e., HCAL and ECAL. Moreover, due to the high-energy nature of the
LHC, a sizable fraction of high-energy photons are produced. The ALPs from
such photons are significantly boosted and thus they can be long-lived enough
to reach the closely located detector modules despite fairly large values of
$m_{a}$ and $g_{a\gamma\gamma}$.

It is remarkable that the existing Run2 data collected with the zero-bias
trigger (red line) allows for sensitivity into the unexplored regions of
parameter space. Adding Run3 data with all high-level triggers (blue line)
further extends the regions that can be probed. Regarding Run4, as mentioned
earlier, we assume that dedicated trigger algorithms are implemented to
capture the full signal flux although designing the triggers is beyond the
scope of this letter. As expected, larger luminosity and acceptance
significantly (green line) improve the sensitivity reaches and expand
explorable regions, not suffering from the beam-dump ``ceiling'' as predicted
earlier. Finally, Run4 data collected by ME0 (brown line) allows us to explore
the large-mass regime above $m_{a}\sim 1$ GeV. The ME0 module will be placed
in the endcap region so that it will accept more energetic signals stemming
from higher-energetic photons, as also suggested by the energy-angle
correlations shown in FIG. 1. We further emphasize that the smallness of the
ME0 module facilitates the implementation of dedicated triggers as they are
less likely to affect the total trigger rate.

Discussion and conclusions. In this letter, we proposed a paradigm-shifting
idea about performing a beam-dump-like measurement at the two LHC general-
purpose detectors, ATLAS and CMS. Under the proposed measurement scheme, the
two calorimeters, HCAL and ECAL, can behave as dumps to which hadrons and
electromagnetic objects inside jets produced by high-energy proton collisions
are dumped. An enormous number of secondary particles including photons are
produced therein while the aforementioned jet constituents are being absorbed.
For illustration purposes, we investigated ALP production via the Primakoff
process of the secondary photons and detection prospects at the muon system of
CMS. In this regard, we carefully assessed the feasibility of DT chambers and
ME0 as signal ``detectors'', identifying the expected experimental signatures
and related potential backgrounds. Our sensitivity study suggests that CMS is
capable of probing a broad range of unexplored parameter space even with the
existing Run2 data collected with the zero-bias trigger.

Our study is predicated upon a few simple assumptions and approximations.
While we performed a simplified GEANT-based simulation study to assess the
possibility to utilize DT chambers and ME0 as signal detectors, a more
dedicated simulation study in line with careful hardware-level consideration
is highly encouraged to assess the feasibility of the proposed ``beam-dump''
measurement at the LHC and identify and estimate related backgrounds more
precisely. Along the line, we expect that signal-background identification
will benefit from the machine-learning-based techniques, as it is deeply
related to pattern recognition. A trigger is another important ingredient to
enhance the signal rate to be recorded. In preparation for future Run4, we
strongly encourage experimental collaborations to develop dedicated trigger
algorithms.

Finally, we emphasize that similar ideas are applicable to other new physics
scenarios, e.g., ALPs interacting with leptons and models of dark photons
decaying into leptons which we plan to investigate in a forthcoming
publication. Also, this analysis in the context of light mediators (vectors
and scalars) can be extended to investigate the parameter spaces associated
with various anomalies, e.g., ATOMKI [41], $g_{\mu}-2$ [42], etc. Using this
new scheme, we hope that ATLAS and CMS will discover new physics in the near
future.

## Acknowledgments

We thank Carlo Battilana, Francesca Romana Cavallo, Karri DiPetrillo, Teruki
Kamon, and Jason Sang Hun Lee for useful discussions. This article was
supported by the computing resources of the Global Science Experimental Data
Hub Center (GSDC) at the Korea Institute of Science and Technology Information
(KISTI). The work of BD, DK, and HK is supported by the U.S. Department of
Energy Grant DE-SC0010813.

## Supplement

Sensitivity in the delayed-decay regime. If the (laboratory-frame) lifetime of
a produced ALP is long enough, $L,\Delta L\ll\tilde{l}$ holds. The detection
probability of the ALP traveling toward the detection module of interest is
approximated to

$P_{\rm det}\approx\frac{\Delta L}{\tilde{l}}\propto g_{a\gamma\gamma}^{2}.$
(8)

Since the ALP production rate is also proportional to $g_{a\gamma\gamma}^{2}$
for any small $g_{a\gamma\gamma}$, if a $x$-times larger data collection were
expected, the new reach $g_{a\gamma\gamma}^{\prime}$ would be approximately

$\displaystyle g_{a\gamma\gamma}^{\prime}$ $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{l
l}\dfrac{g_{a\gamma\gamma}}{x^{1/4}}&\hbox{for zero
backgrounds,}\\\\[10.00002pt] \dfrac{g_{a\gamma\gamma}}{x^{1/8}}&\hbox{for
large backgrounds,}\end{array}\right.$ (11)

where we determine $N_{\rm sig}$, which is required to have the sensitivity,
with statistical error only. Therefore, unlike the case of $L,\Delta
L\gg\tilde{l}$, the increase of data allows to continue the exploration of
more parameter space toward the delayed-decay regime, i.e., smaller $m_{a}$
and $g_{a\gamma\gamma}$.

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