11institutetext: ARC Centre of Excellence for Particle Physics at the
Terascale, School of Physics, University of Melbourne, 3010,
Australia22institutetext: Department of Physics and Astronomy, University of
British Columbia, Vancouver, British Columbia, V6T 1Z1 Canada33institutetext:
Institute of Particle Physics, Vancouver, British Columbia, V6T 1Z1
Canada44institutetext: DESY, Notkestrasse 85, D-22607 Hamburg,
Germany55institutetext: Institute for Theoretical Particle Physics and
Cosmology (TTK), RWTH Aachen University, D-52056 Aachen, Germany

# Revised constraints and Belle II sensitivity for visible and invisible
axion-like particles

Matthew J. Dolan 2 Torben Ferber 2,3 Christopher Hearty 4,5 Felix Kahlhoefer 4
and Kai Schmidt-Hoberg<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Light pseudoscalars interacting pre-dominantly with Standard Model gauge
bosons (so-called axion-like particles or ALPs) occur frequently in extensions
of the Standard Model. In this work we review and update existing constraints
on ALPs in the keV to GeV mass region from colliders, beam dump experiments
and astrophysics. We furthermore provide a detailed calculation of the
expected sensitivity of Belle II, which can search for visibly and invisibly
decaying ALPs, as well as long-lived ALPs. The Belle II sensitivity is found
to be substantially better than previously estimated, covering wide ranges of
relevant parameter space. In particular, Belle II can explore an interesting
class of dark matter models, in which ALPs mediate the interactions between
the Standard Model and dark matter. In these models, the relic abundance can
be set via resonant freeze-out, leading to a highly predictive scenario
consistent with all existing constraints but testable with single-photon
searches at Belle II in the near future.

###### Keywords:

Mostly Weak Interactions: Beyond Standard Model; Collider Physics:
$e^{+}$-$e^{-}$ Experiments; Astroparticles: Cosmology of Theories beyond the
SM

††preprint: DESY-17-127

##  1 Introduction

Axions and axion-like particles (ALPs) are a generic feature of many
extensions of the Standard Model (SM), occurring for example in most solutions
of the strong CP problem (see refs. Hook:2014cda ; Fukuda:2015ana  for recent
work in this direction), in string compactifications Arvanitaki:2009fg ;
Cicoli:2012sz  and in models with broken supersymmetry (the so-called R-axion
Bellazzini:2017neg ). Being Pseudo-Goldstone bosons, they can naturally be
light and very weakly coupled, thus evading many of the strong constraints on
new physics imposed for example by the LHC. In fact, the most promising way to
search for ALPs may be the intensity frontier Hewett:2012ns ; Essig:2013lka ,
i.e. searches with relatively low energy but very large integrated
luminosity/intensity. The coming years promise significant progress in this
area, driven first of all by the upcoming Belle II experiment Abe:2010gxa ,
but also for example by NA62 NA62:2017rwk  and by longer-term projects such as
the planned SHiP facility Anelli:2015pba . These searches may soon shed light
on the existence and nature of ALPs.

ALPs with masses below the MeV scale can have a wide range of implications for
cosmology and astrophysics Cadamuro:2011fd , affecting for example Big Bang
Nucleosynthesis (BBN) Millea:2015qra , the Cosmic Microwave Background (CMB)
and the evolution of stars. ALPs can constitute cold dark matter (DM)
Arias:2012az  and have also been considered as possible explanations for a
range of astrophysical anomalies, such as the over-efficient cooling of
certain classes of stars Giannotti:2015kwo , the surprising transparency of
the Universe to very high-energy $\gamma$-rays Meyer:2013pny  or the hints for
a mono-energetic x-ray line around 3.5 keV Cicoli:2014bfa ; Conlon:2014xsa ;
Jaeckel:2014qea .

Over the past few years an increasing amount of interest has been paid to ALPs
in the MeV to GeV range Mimasu:2014nea ; Dolan:2014ska ; Jaeckel:2015jla ;
Dobrich:2015jyk ; Izaguirre:2016dfi ; Knapen:2016moh ; Brivio:2017ije ;
Bauer:2017nlg ; Choi:2017gpf ; Bauer:2017ris . In this mass range, ALPs are
largely irrelevant for astrophysics and cosmology, but they can have a number
of interesting implications for particle physics. For example, ALPs have been
considered as an explanation for the anomalous magnetic moment of the muon
Chang:2000ii ; Marciano:2016yhf ; Bauer:2017nlg  or for exotic resonances in
nuclear transitions Ellwanger:2016wfe . Moreover, it has been pointed out that
ALPs may play a crucial role in electroweak symmetry breaking and the solution
of the hierarchy problem Flacke:2016szy  via the so-called relaxion mechanism
Graham:2015cka . Finally, ALPs offer an interesting possibility to connect the
SM to a potential DM particle in such a way that thermal freeze-out can be
reconciled with constraints from direct detection experiments Nomura:2008ru ;
Boehm:2014hva ; Dolan:2014ska .

Given the level of detail and sophistication of many recent analyses, it is
rather surprising that many constraints on the ALP parameter space have not
been updated for a very long time. In fact, a number of constraints shown in
recent studies are taken directly from the very early work by Massó and Toldrà
Masso:1995tw ; Masso:1997ru , even though a wealth of newer experimental data
has since become available. Moreover, the early calculations were never meant
to be more than order-of-magnitude estimates and are therefore potentially
misleading when compared to the projected sensitivities of future experiments.
The aim of this paper is to revisit constraints on couplings of ALPs to
photons and hypercharge gauge bosons for ALP masses in the MeV to GeV range
and to compare existing constraints to the sensitivity of upcoming searches
for ALPs. For this purpose, we will discuss bounds from electron-positron
colliders, from electron beam dumps and from Supernova (SN)
1987A.111Constraints on ALPs from proton beam dump experiments have been
studied in detail in ref. Dobrich:2015jyk . While many of these constraints
have been investigated in detail in the context of different models, such as
hidden photons or light scalars (see e.g. Batell:2009di ; Andreas:2012mt ;
Essig:2013vha ; Izaguirre:2013uxa ; Batell:2014mga ; Krnjaic:2015mbs ), we
present the first comprehensive reinterpretation of these constraints in the
context of ALPs.

A central part of our work is to calculate in detail the projected sensitivity
to ALPs of Belle II. While the primary purpose of Belle II is to study the
properties of B-mesons, the experiment is in fact ideally suited for a wide
range of new-physics searches. Several previous studies have investigated the
potential of Belle II to search for invisible Dark Photon decays Essig:2013vha
; Izaguirre:2013uxa ; lit:b2tip , which will significantly extend the range of
earlier searches at BaBar Aubert:2008as ; delAmoSanchez:2010ac ; Lees:2017lec
. Here we present the first realistic study of experimental backgrounds and
detection efficiencies for ALP searches at Belle II. We point out that
previous studies have significantly underestimated the sensitivity of Belle II
for single-photon searches due to overly conservative background estimates
based on an extrapolation of BaBar results. Our study demonstrates that Belle
II can already explore new parameter space with early data and in the long run
will be highly complementary to future searches for ALPs at the LHC
Bauer:2017ris  or at SHiP Alekhin:2015byh ; Dobrich:2015jyk . We conclude that
Belle II therefore possesses a unique opportunity to discover both visibly and
invisibly decaying ALPs.

The latter case is of particular interest in the context of models where the
ALP is responsible for mediating the interactions between DM and SM particles.
We focus on the case that DM couples pre-dominantly to photons. Strong
constraints on this scenario from the CMB and indirect detection experiments
can be avoided if DM annihilations during freeze-out are resonantly enhanced.
We find that in this scenario the observed DM relic abundance can be
reproduced in a well-defined region of parameter space, which is presently
allowed by all experimental constraints but can be largely probed in the next
few years with single-photon searches at Belle II.

This paper is structured as follows. Section 2 provides a general review of
the effective interactions of ALPs and establishes the notation used in the
rest of this paper. The focus of section 3 is on the discussion of existing
constraints on the ALP parameter space. The special case of an ALP coupled to
DM is discussed in section 4. Finally, we present the projected sensitivity of
Belle II in section 5, before concluding in section 6.

##  2 Effective interactions of ALPs

This work focuses on the interactions of a pseudoscalar ALP $a$ with SM gauge
bosons. Specifically, we consider the Lagrangian

$\mathcal{L}=\frac{1}{2}\partial^{\mu}a\,\partial_{\mu}a-\frac{1}{2}m^{2}_{a}\,a^{2}-\frac{c_{B}}{4\,f_{a}}\,a\,B^{\mu\nu}\tilde{B}_{\mu\nu}-\frac{c_{W}}{4\,f_{a}}\,a\,W^{i,\mu\nu}\tilde{W}^{i}_{\mu\nu}\;,$
(1)

where $B^{\mu\nu}$ and $W^{i,\mu\nu}$ denote the field strength of $U(1)_{Y}$
and $SU(2)_{L}$, respectively, and we have defined the dual field strength
tensors via
$\tilde{B}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\,B^{\rho\sigma}$.
The parameters $m_{a}$ and $f_{a}$ denote the ALP mass and decay constant,
which we assume to be independent parameters.

We emphasize that the Lagrangian that we consider does not include all terms
that would be expected to be present in a general effective field theory
description of ALPs Brivio:2017ije ; Bauer:2017ris . In particular, we do not
consider interactions between ALPs and SM fermions or interactions between
ALPs and gluons. This restriction is well-motivated in models where the
interactions between ALPs and the SM arise from new heavy fermions that do not
carry colour charge. The reason we are interested in ALPs that do not couple
to gluons and fermions is that such interactions typically lead to flavour-
changing processes (via penguin diagrams or via mixing with the $\pi^{0}$),
which are tightly constrained by searches for rare decays Dolan:2014ska ;
Choi:2017gpf . ALPs coupling dominantly to the gauge bosons of $U(1)_{Y}$ and
$SU(2)_{L}$, on the other hand, are much harder to probe experimentally and
require dedicated experimental search strategies.222We note that from the
interactions that we consider couplings to fermions will be introduced at the
one-loop level and couplings to gluons will appear at the two-loop level.
These couplings are however too small to have relevant experimental
consequences Bauer:2017ris .

Many of the constraints that we discuss below will remain valid even if the
ALP has additional interactions. In fact, additional interactions are expected
to increase the ALP production cross section and hence lead to even stronger
bounds. Nevertheless, in some cases the presence of additional interactions
may actually weaken the constraints. For example, additional decay modes will
_decrease_ the ALP lifetime and therefore potentially suppress constraints
from experiments searching for long-lived particles. Additional interactions
may also lead to the trapping of ALPs in astrophysical objects, weakening the
constraints obtained from such systems. The reader should therefore be careful
when applying the bounds presented in this work to more complicated models.

After electroweak symmetry breaking, the two terms in eq. (1) induce four
different interactions between the ALP and SM gauge bosons:

$\mathcal{L}\supset-\frac{g_{a\gamma\gamma}}{4}aF_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{g_{a\gamma
Z}}{4}aF_{\mu\nu}\tilde{Z}^{\mu\nu}-\frac{g_{aZZ}}{4}aZ_{\mu\nu}\tilde{Z}^{\mu\nu}-\frac{g_{aWW}}{4}aW_{\mu\nu}\tilde{W}^{\mu\nu}\;,$
(2)

where the field strengths and their duals are defined as above. The individual
couplings can be calculated in terms of the parameters introduced above. The
two couplings of greatest interest for the purpose of this work are

$g_{a\gamma\gamma}=\frac{c_{B}\,\cos^{2}\theta_{\mathrm{W}}+c_{W}\,\sin^{2}\theta_{\mathrm{W}}}{f_{a}}\,,\qquad
g_{a\gamma Z}=\frac{\sin 2\theta_{\mathrm{W}}(c_{W}-c_{B})}{f_{a}}\;,$ (3)

where $\theta_{\mathrm{W}}$ denotes the Weinberg angle.

If $c_{B}$ and $c_{W}$ are independent parameters, so are $g_{a\gamma\gamma}$
and $g_{a\gamma Z}$. In particular, for $c_{B}\approx c_{W}$ one finds
$g_{a\gamma Z}\ll g_{a\gamma\gamma}$. Nevertheless, as pointed out in ref.
Izaguirre:2016dfi , there are potentially strong constraints on $c_{W}$ from
loop-induced flavour-changing processes like $B\to Ka$. It is therefore
particularly interesting to consider the case where $c_{W}\ll c_{B}$ and hence

$g_{a\gamma\gamma}\approx-\frac{1}{2}\cot\theta_{\mathrm{W}}g_{a\gamma
Z}\approx-0.94g_{a\gamma Z}\;.$ (4)

We will refer to the case $c_{B}\sim c_{W}$ (and hence $g_{a\gamma Z}\ll
g_{a\gamma\gamma}$) as _photon coupling_ and to the case $c_{B}\gg c_{W}$ (and
hence $g_{a\gamma Z}\sim-g_{a\gamma\gamma}$) as _hypercharge coupling_. We
emphasize that in both cases ALPs will also couple to pairs of heavy gauge
bosons, but we do not discuss the effect of these couplings further.

The $a\gamma\gamma$-interaction is of particular importance, as it determines
the lifetime $\tau_{a}$ of the ALP. The decay width $\Gamma_{a}=\tau_{a}^{-1}$
is given by

$\Gamma_{a}=\frac{g_{a\gamma\gamma}^{2}\,m_{a}^{3}}{64\pi}\;.$ (5)

It is worth emphasizing that for $g_{a\gamma\gamma}\ll 1\,\mathrm{TeV^{-1}}$
and for $m_{a}\ll 1\,\mathrm{GeV}$ this decay width is extremely small and
hence the ALP decay length can be very large, in particular if the ALPs are
produced with significant boost $\gamma_{a}=E_{a}/m_{a}$. For a detector of
size $L_{\mathrm{D}}$ the fraction of ALPs that decay within the detector is
given by333This expression assumes that either all ALPs are produced
approximately in the same direction (as in beam dump experiments) or that the
detector is approximately spherical. In all other cases both $L_{\mathrm{D}}$
and $p_{a}$ depend on the direction of the ALP momentum Bauer:2017ris .

$p_{a}=1-\exp\left(-\frac{L_{\mathrm{D}}}{\gamma_{a}\,\tau_{a}}\right)\;.$ (6)

If the ALPs escape from the detector before decaying, this can prohibit
searches for the decay $a\to\gamma\gamma$. On the other hand, such long decay
lengths can facilitate a different kind of search, which focuses on the
missing momentum carried away by the invisible ALP. We will discuss existing
results and future prospects for both search strategies below.

Figure 1:  Feynman diagrams for ALP production in $e^{+}e^{-}$ collisions via
ALP-strahlung (left) and photon fusion (right) and the subsequent decay of the
ALP into two photons.

The same interaction can also be responsible for the production of ALPs, for
example in $e^{+}e^{-}$ collisions. There are two different production
processes of interest: ALP-strahlung ($e^{+}e^{-}\to\gamma^{\ast}\to\gamma+a$)
and photon fusion ($e^{+}e^{-}\to e^{+}e^{-}+a$), see figure 1. For the former
process (and in the limit $m_{a}\to 0$) the differential cross section with
respect to the photon angle in the centre-of-mass (CM) frame is given by
Marciano:2016yhf

$\frac{\mathrm{d}\sigma}{\mathrm{d}\cos\theta}=\frac{g_{a\gamma\gamma}^{2}\,\alpha}{128}(3+\cos
2\theta)(1-m_{a}^{2}/s)^{3}\;,$ (7)

which has a mild angular dependence and is notably independent of the CM
energy $\sqrt{s}$ for $m_{a}\ll\sqrt{s}$.444Even for very light ALPs there
remains a slight dependence on $\sqrt{s}$ due to the running of both $\alpha$
and $g_{a\gamma\gamma}$, which can change by up to 10% over the range of
energies that we consider Bauer:2017ris . ALP-strahlung therefore typically
leads to a photon with sizeable transverse momentum, which is a promising
experimental signature.

The cross section for ALP production via photon fusion can be calculated by
replacing the colliding particles by their equivalent photon spectra
$\gamma(x)$ and making use of the ALP production cross section from a pair of
photons Dobrich:2015jyk :

$\sigma(\gamma\gamma\rightarrow
a)=\frac{\pi\,g_{a\gamma\gamma}^{2}\,m_{a}}{16}\delta(m_{\gamma\gamma}-m_{a})\;.$
(8)

Unless $m_{a}$ is close to $\sqrt{s}$, ALP production via photon fusion
typically dominates over ALP-strahlung. However, the ALPs produced in this way
are much harder to detect experimentally, as they carry only little energy and
therefore decay into relatively soft photons in the laboratory frame. We will
return to the experimental feasibility of searches for ALPs produced in photon
fusion in section 5.3.

This work focuses on ALPs with mass below $10\,\mathrm{GeV}$, so that the
decay $a\to\gamma Z$ is forbidden. The $a\gamma Z$ interaction nevertheless
plays an important role, as it leads to the decay $Z\to\gamma+a$ Dror:2017nsg
; Bauer:2017ris  with partial decay width given by

$\Gamma(Z\to\gamma+a)=\frac{g_{a\gamma
Z}^{2}}{384\pi}\left(\frac{m_{Z}^{2}-m_{a}^{2}}{m_{Z}}\right)^{3}\;.$ (9)

Depending on the ALP lifetime, this process can either lead to the signature
$Z\to\gamma+\text{inv}$ or to $Z\to 3\gamma$, both of which can be tightly
constrained by experiments.

To conclude this section we note that in principle ALPs may also be produced
in Higgs decays, $h\to Za$ or $h\to aa$, leading to strong constraints from
the non-observation of these decay modes Bauer:2017nlg ; Bauer:2017ris . These
interactions however only appear when considering effective operators of
dimension 6 or higher, and they are not directly linked to the interactions
between ALPs and SM gauge bosons. While it is instructive to include these
interactions in a general effective field theory approach, they are not
generic and may be absent in specific UV completions. We will therefore not
consider exotic Higgs decays in this work and instead focus on the
phenomenology of the interactions between ALPs and gauge bosons.

##  3 Review of bounds on the ALP parameter space

In this section we review existing bounds on the ALP parameter space, updating
constraints wherever new data or more precise calculations have become
available. Most of the constraints that we will discuss only probe the
effective ALP-photon coupling $g_{a\gamma\gamma}$. The only exception are
constraints from high-energy colliders, which depend on whether the ALP
couples to photons or hypercharge. We show a summary of all relevant
constraints for both cases in figure 2. All collider and beam dump bounds are
provided at $95\%$ confidence level (CL), with the exception of the bounds
from BaBar, which are provided at $90\%$ CL. Given that the parameter space
under consideration covers many orders of magnitude, the difference between
the two choices of CL is imperceptible.

Figure 2:  Existing constraints on ALPs with photon coupling (left) and
hypercharge coupling (right). Proton beam dump constraints are taken from ref.
Dobrich:2015jyk , LEP constraints on $e^{+}e^{-}\to\gamma\gamma$ from ref.
Jaeckel:2015jla , CDF constraints on $Z\to\gamma\gamma$ from ref.
Bauer:2017ris , bounds from horizontal branch stars from ref. Cadamuro:2011fd
, bounds from visible decays of ALPs produced in SN 1987A from ref
Jaeckel:2017tud  and bounds from heavy-ion collisions from ref. Knapen:2017ebd
. All other constraints have been revisited and updated in the present work.

###  3.1 Bounds from electron-positron colliders

#### Mono-photon searches at LEP.

Relevant bounds on the ALP parameter space are obtained from so-called mono-
photon searches, i.e. searches for highly-energetic photons in association
with missing energy resulting from the process
$e^{+}e^{-}\to\gamma^{\ast}\to\gamma+a(\to\text{inv})$. The bound
conventionally shown in the ALP parameter space is often attributed to LEP
(see e.g. ref. Mimasu:2014nea ), but actually goes back to the early analysis
from ref. Masso:1995tw  of a mono-photon search at the ASP experiment at SLAC
Hearty:1989pq . Upon closer inspection, however, it turns out that this search
is not sensitive to ALP masses in the sub-GeV region, because it requires
$E_{\gamma}<10\,\text{GeV}$ in the fiducial region, whereas for light ALPs the
photon energy is given by $E_{\gamma}\approx\sqrt{s}/2=14.5\,\mathrm{GeV}$.

To determine the sensitivity of LEP for ALPs, we follow the re-analysis from
ref. Fox:2011fx  of a mono-photon search at DELPHI Abdallah:2008aa  based on
$650\,\mathrm{pb^{-1}}$ at CM energies between 180 GeV and 209 GeV. The best
sensitivity for an ALP signal stems from the High Density Projection Chamber,
which covers $45^{\circ}<\theta<135^{\circ}$, and from the three highest-
energy bins included in the analysis,
$0.9\,E_{\text{beam}}<E_{\gamma}<1.05\,E_{\text{beam}}$. We implement detector
efficiencies and resolution as detailed in ref. Fox:2011fx  and assume that an
ALP will escape unnoticed if it travels more than
$L_{\mathrm{D}}=260\,\mathrm{cm}$ in the radial direction without decaying
Aarnio:1990vx .555The event selection includes a veto on energy depositions in
any electromagnetic calorimeter. We hence approximate the detector length by
the outer radius of the High Density Projection Chamber.

We note that mono-photon searches have also been carried out at the LHC (for
the most recent analyses see refs. Aaboud:2017dor ; CMS:2017ysu ), but their
sensitivity does not significantly improve on the bound from LEP
Mimasu:2014nea . Moreover, for these searches the validity of the ALP
effective theory becomes a concern Mimasu:2014nea ; Brivio:2017ije . We
therefore do not show bounds from LHC mono-photon searches.

#### Radiative Upsilon decays.

A related class of searches, which can be performed at $B$-factories, are
searches for radiative decays of $\Upsilon(nS)$ with $n=1,2,3$. While these
searches are typically interpreted in terms of new invisible particles
coupling to $b$-quarks, they also apply to the case where the new particle
couples directly to the photon: $\Upsilon(nS)\to\gamma^{\ast}\to\gamma+a$. In
fact, the corresponding branching ratio is easily calculated in terms of the
branching ratio into electrons Masso:1995tw :

$\text{BR}(\Upsilon(nS)\to\gamma+a)=\frac{\alpha\,g_{a\gamma\gamma}^{2}\,m_{b}^{2}}{8\pi}\cdot\text{BR}(\Upsilon(nS)\to
e^{+}e^{-})\;,$ (10)

where $m_{b}$ denotes the $b$-quark mass.

The bound conventionally shown comes from the Crystal Ball experiment, which
gives $\text{BR}(\Upsilon(1S)\to\gamma+\text{inv})<4.0\times 10^{-5}$
Antreasyan:1990cf . However, much stronger bounds can be obtained from more
recent measurements, such as the bound
$\text{BR}(\Upsilon(3S)\to\gamma+\text{inv})<3\times 10^{-6}$ from BaBar
Aubert:2008as . Here we reinterpret this latter constraint under the
assumption that the ALP will escape from the detector, if it travels a
distance of $L_{\mathrm{D}}=275$ cm from the interaction point without
decaying.666The event selection includes a veto of energy depositions in the
instrumented flux return (IFR). We assume that a photon has sizeable detection
efficiency only if it interacts before it reaches the two outermost IFR
absorber layers.

#### Dark Photon searches at BaBar.

BaBar has published results from a search for invisibly decaying Dark Photons
produced in association with an ordinary photon, $e^{+}e^{-}\to\gamma
A^{\prime}(\to\text{inv})$ Lees:2017lec . This search differs from an
analogous ALP search in that the photon distribution peaks strongly at small
polar angles $\theta$ in the CM frame. In the limit $m_{A^{\prime}}\to 0$ one
finds Izaguirre:2013uxa

$\frac{\mathrm{d}\sigma}{\mathrm{d}\cos\theta}=\frac{(1+\cos^{2}\theta)\pi\,\alpha^{2}\,\epsilon^{2}}{2\,\sin^{2}\theta\,E_{\text{beam}}^{2}}\;,$
(11)

where $\epsilon$ denotes the kinetic mixing parameter.777Note that this
expression at face value implies a divergent total cross section. In a more
accurate treatment this is regulated by the finite electron mass, which we
have neglected. Since realistic detectors will not be sensitive to photon
angles close to $0$ or $\pi$, these details do not matter. To convert a bound
on $\epsilon$ for Dark Photons into a bound on $g_{a\gamma\gamma}$ for ALPs we
therefore have to correct for the fact that the geometric acceptance will be
very different in the two cases.

The BaBar analysis considers $-0.6<\cos\theta<0.6$ for
$m_{A^{\prime}}>5.5\,\text{GeV}$ and $-0.4<\cos\theta<0.6$ for
$m_{A^{\prime}}<5.5\,\text{GeV}$. By integrating the respective differential
cross sections for ALP production and Dark Photon production over these ranges
we obtain the fiducial cross section including geometric acceptance. Using
these numbers, we can translate bounds on Dark Photons into the ALP parameter
space under the assumption that all other selection cuts have the same
efficiency for the two models. For very small masses of the invisibly decaying
particle, we find that the translation is given by

$g_{a\gamma\gamma}=1.8\times
10^{-4}\,\text{GeV}^{-1}\left(\frac{\epsilon}{10^{-3}}\right)\;.$ (12)

Repeating this calculation for finite ALP masses and taking into account the
probability that the ALP decays before leaving the detector (see above) using
a detector length of $L_{\mathrm{D}}=275$ cm Aubert:2001tu , we can then
reinterpret the full BaBar bound in the context of ALPs.

#### Radiative Z-boson decays.

If $g_{a\gamma Z}$ is non-zero, ALPs can also be produced in the decay
$Z\to\gamma+a$. The resulting experimental signature depends on the ALP decay
length. If the decay length is large compared to the size of the detector, the
most promising search channel is $Z\to\gamma+\text{inv}$. This process has for
example been studied by the L3 collaboration at LEP Acciarri:1997im , which
quotes an upper limit on the corresponding branching ratio of
$\text{BR}(Z\to\gamma+a)<1.1\times 10^{-6}$. We reinterpret this bound under
the assumption that the ALP escapes if it does not decay within
$L_{\mathrm{D}}=180\,\mathrm{cm}$ from the interaction point L3:1989aa .888The
event selection includes a veto of energy depositions in any calorimeter. We
hence approximate the detector length by the outer radius of the barrel
hadronic calorimeter.

Constraints of similar strength are obtained for ALPs decaying close to the
interaction point. If the ALP has a high boost but a short lifetime, the two
photons produced in its decay will be approximately collinear and may thus
appear in the detector as a single photon. The resulting signature can then
mimic the forbidden decay $Z\to 2\gamma$ Jaeckel:2015jla ; Chala:2015cev .
Constraints on this decay mode from LEP have been discussed in ref.
Jaeckel:2015jla , but ref. Bauer:2017ris  points out that for $m_{a}\lesssim
m_{\pi^{0}}$ even stronger constraints can be derived from the CDF result
$\text{BR}(Z\to\gamma\gamma)<1.45\times 10^{-5}$ Aaltonen:2013mfa .

For $m_{a}\gtrsim 10\,\mathrm{GeV}$, on the other hand, the photons from the
ALP decay can easily be distinguished and one obtains the signature $Z\to
3\gamma$. A recent ATLAS search constrains this decay mode to $\text{BR}(Z\to
3\gamma)<2.2\times 10^{-6}$ Aad:2015bua . Future LHC searches are expected to
significantly improve this bound and to extend the sensitivity to lower ALP
masses Bauer:2017ris .

###  3.2 Bounds from beam dump experiments

ALPs can be produced in electron and proton beam dump experiments via
Primakoff production, i.e. the conversion of a photon into an ALP in the
vicinity of a nucleus Tsai:1986tx . A number of relevant electron beam dump
experiments have been studied in ref. Bjorken:2009mm , while constraints from
proton beam dump experiments have been reviewed in ref. Dobrich:2015jyk .

#### SLAC E141.

Although E141 primarily searched for long-lived particles decaying to
$e^{+}e^{-}$ Riordan:1987aw , during some of the data taking a photon
converter was inserted in front of the detector, giving the experiment
sensitivity also for ALPs decaying to photons. The results from this search
have been presented in ref. Krasny ; Dobrich:2017gcm . Although results are
only provided for a limited range in ALP masses, the given details on the
detector geometry are sufficient to extend the search range by rescaling with
the appropriate decay probability and assuming that the positron energy
spectrum is independent of the ALP mass as long as $m_{a}\ll E_{\text{beam}}$.

To perform the rescaling of the decay probability, we need to make an
assumption on the typical energy of ALPs that leads to an observable signal,
i.e. a positron with $E>E_{\text{beam}}/2$. We find that good agreement with
ref. Krasny  is obtained for $E_{a}\approx 6.5\,\text{GeV}$. Previous attempts
to reinterpret data from E141 have assumed that smaller ALP energies are
sufficient to produce sufficiently highly-energetic positrons, leading to
somewhat more aggressive bounds Hewett:2012ns .

#### SLAC E137.

The E137 experiment at SLAC has published results from a dedicated search for
ALPs coupling only to photons Bjorken:1988as . The paper does not consider the
turnover of the exclusion limit towards large couplings due to the exponential
suppression of the number of ALPs that reach the detector, but it is possible
to include this additional effect with the information provided in the paper.
Specifically, we start from the photon track-length distribution as a function
of energy provided in figure 14 of ref. Bjorken:1988as  and assume that the
cooling water in the beam dump yields the dominant contribution to the ALP
production.999Assuming the dominant production to come from the aluminium
plates does not significantly change our results. While the production rate
for aluminium is larger than that for oxygen by around 20%, this is
compensated by aluminium having a slightly larger form factor suppression.

Although the total decay length between absorber and detector is
$204\,\mathrm{m}$, ALPs decaying at the beginning of the decay volume will
often not lead to an observable signal, since the photons produced in the
decay may miss the detector, which only has a radius $r_{\mathrm{D}}\approx
1.5\,\mathrm{m}$. The typical opening angle between the two photons produced
in the ALP decay is $\theta_{\gamma\gamma}\sim 2/\gamma_{a}=2\,m_{a}/E_{a}$.
If the ALP decay happens at a distance $\Delta z$ from the detector, the two
photons will thus hit the detector with a typical separation of
$\theta_{\gamma\gamma}\Delta z$. To be conservative, we therefore assume that
an ALP decay will only lead to an observable signal in the detector if $\Delta
z<E_{a}\,r_{\mathrm{D}}/m_{a}$, so that the effective length of the decay
volume is reduced for ALPs with small boost factor. This procedure allows us
to extend the bound from ref Bjorken:1988as  to the full parameter space
relevant for ALP searches. Our final result roughly agrees with the one shown
in ref. Hewett:2012ns , although we find slightly stronger constraints towards
larger couplings by using a more realistic photon distribution in our
calculation.

#### Proton beam dumps.

Constraints on the ALP parameter space from proton beam dumps have recently
been studied in ref. Dobrich:2015jyk . Here we show the bounds obtained in
that work from CHARM Bergsma:1985qz  and NuCal Blumlein:1990ay ;
Blumlein:1991xh .

###  3.3 Bounds from heavy-ion collisions

Ref. Knapen:2016moh  proposed a novel way to search for ALPs in
ultraperipheral heavy-ion collisions. In such collisions ALPs can be produced
via the fusion of two coherently emitted photons, i.e. via diagrams analogous
to the one shown in the right panel of figure 1. The resulting ALPs have very
low boost factors and their decays consequently lead to two very soft photons
that are approximately back-to-back. The minimum photon energy required by the
trigger places a lower bound on the ALP masses that can be probed with this
search strategy. A reinterpretation of the recent ATLAS search for light-by-
light scattering Aaboud:2017bwk  leads to relevant constraints for ALPs with
$m_{a}>7\,\mathrm{GeV}$ Knapen:2017ebd .

###  3.4 Bounds from astrophysics

#### Supernova 1987A.

Weakly coupled particles such as axions or ALPs with masses up to about
$100\,\text{MeV}$ can be copiously produced in the hot core of a supernova.
Because of their weak couplings these particles stream out of the core and
thereby constitute a new energy loss mechanism. In the absence of such new
particles the main cooling mechanism is due to neutrino emission. The
corresponding neutrino signal has been observed in the case of SN 1987A,
placing a bound on possible exotic energy loss mechanisms, which should not
exceed the energy loss via neutrino emission.

ALPs that couple exclusively to photons are produced via the Primakoff
process. As electrons are highly degenerate in the supernova core, their phase
space is Pauli-blocked and their contribution to ALP production is negligible.
Protons are only partially degenerate and correspondingly the process
$\gamma+p\rightarrow p+a$ is the main production mode. The resulting ALP
energy spectrum has been calculated with detailed account of the production
process in a core-collapse supernova Payez:2014xsa . An analytical fit of
these results has been performed in ref. Jaeckel:2017tud , which yields an
approximate expression for the ALP production rate:

$\frac{\mathrm{d}N_{a}}{\mathrm{d}E_{a}}\sim\frac{C\,E_{a}^{2}\,\sigma_{\text{pr}}(E_{a})}{\exp(E_{a}/T)-1}\;,$
(13)

where $T$ is the effective temperature and $\sigma_{\text{pr}}(E_{a})$ is the
cross section for Primakoff production Cadamuro:2011fd . For SN 1987A a good
fit is obtained for $C=2.54\times 10^{77}\,\mathrm{MeV^{-1}}$ and
$T=30.6\,\mathrm{MeV}$. The total energy outflow is then given by

$E=\int_{m_{a}}^{\infty}\frac{\mathrm{d}N_{a}}{\mathrm{d}E_{a}}E_{a}\,\mathrm{d}E_{a}\;,$
(14)

which should be smaller than $\sim 3\times 10^{53}\,\text{erg}$. Using this
constraint leads to an upper bound on the ALP coupling of around
$g_{a\gamma\gamma}<6\times 10^{-9}\,\text{GeV}^{-1}$ for small ALP masses.

This bound however does not apply for arbitrarily large couplings, because at
some point the axions will interact so strongly that they are trapped in the
supernova core. The ALP mean free path is given by

$\lambda_{a}=\frac{1}{n_{p}\,\sigma_{\text{bc}}}\;,$ (15)

where $n_{p}$ denotes the proton density and $\sigma_{\text{bc}}$ is the cross
section for back-conversion, which is directly related to the production cross
section Brockway:1996yr :

$\sigma_{\text{bc}}\simeq\frac{2}{\beta_{a}^{2}}\sigma_{\text{pr}}(E_{a})$
(16)

with $\beta_{a}=\sqrt{1-m_{a}^{2}/E_{a}^{2}}$. But even if the ALP mean free
path is smaller than the size of the supernova core ($\sim 10\,\text{km}$),
energy transport via ALPs can still be large. The new particle is harmless
only if it interacts more strongly than the particles which provide the
standard mode of energy transfer, i.e. neutrinos.

To estimate the rate of energy transport, we can calculate the ALP Rosseland
mean opacity $\kappa_{a}$. For the inverse Primakoff process this can be
written as Raffelt:1996wa

$\kappa_{a}^{\text{P}}=\frac{\int_{m_{a}}^{\infty}\mathrm{d}E_{a}\,E_{a}^{3}\,\beta_{a}\,\frac{\partial}{\partial
T}\frac{1}{\exp(E_{a}/T)-1}}{\rho\int_{m_{a}}^{\infty}\mathrm{d}E_{a}\,\lambda_{a}\,E_{a}^{3}\,\beta_{a}\frac{\partial}{\partial
T}\frac{1}{\exp(E_{a}/T)-1}}\;,$ (17)

while the contribution from decay is given by Raffelt:1988rx

$\kappa_{a}^{\text{D}}=\frac{(2\pi)^{7/2}}{45\,\rho}\left(\frac{T}{m_{a}}\right)^{5/2}e^{m_{a}/T}\,\Gamma_{a\gamma\gamma}\;.$
(18)

The constraint from SN 1987A only applies if
$\kappa_{a}=\kappa_{a}^{\text{P}}+\kappa_{a}^{\text{D}}<\kappa_{\nu}$, where
$\kappa_{\nu}\approx 8\times 10^{-17}\mathrm{cm^{2}/g}$ is the neutrino
opacity Masso:1995tw . For small ALP masses this corresponds to
$g_{a\gamma\gamma}\lesssim 7\times 10^{-6}\,\mathrm{GeV^{-1}}$.

An additional constraint from SN 1987A can be obtained by considering not only
the energy loss due to ALP emission, but also the visible signal resulting
from the “ALP burst” if the ALPs subsequently decay into photons
Jaeckel:2017tud . In figure 2 we show the constraint obtained in ref.
Jaeckel:2017tud , which extends the bound from SN 1987A to even smaller
couplings.

We emphasize that our treatment of the limit from SN1987A remains somewhat
simplistic. Our parametrisation of the ALP energy spectrum and outflow are
based on a fit to simulations performed with
$g_{a\gamma\gamma}=10^{-10}\mathrm{GeV^{-1}}$ for a light ALP, and it is
unclear how accurate this is over for large values $g_{a\gamma\gamma}$ and for
$m_{a}$ near the supernova core temperature. Modern models of supernovae (see
ref. Janka:2012wk ) are also considerably more sophisticated than ours, which
assumes a fixed nuclear density and temperature (following the treatment in
refs. Raffelt:1987yb ; Masso:1995tw ). The mean free path of neutrinos depends
sensitively on radius, time and energy, making it difficult to encode the
effect in terms of a single opacity Buras:2005rp . We estimate this introduces
a theoretical uncertainty of at least a factor of two into our limits. To
reduce this uncertainty will likely require a dedicated simulation of heat
transfer in the core of a supernova. While some work in this direction has
been undertaken for axions Payez:2014xsa ; Fischer:2016cyd , it would be
interesting to perform dedicated simulations for the ALP scenario.

#### Horizontal branch stars.

Constraints on the ALP parameter space can be obtained by considering the
cooling of horizontal branch (HB) stars Raffelt:1987yb ; Raffelt:1996wa .
These constraints have recently been investigated in ref. Cadamuro:2011fd ,
and we show the bounds derived there. As with supernovae, it would be
interesting to undertake more detailed simulations for ALPs, as has been done
for axions in ref. Friedland:2012hj .

#### Big Bang Nucleosynthesis.

It was pointed out in ref. Cadamuro:2011fd  that the ALP parameter space is
potentially strongly constrained by BBN, if the ALP decay rate is comparable
to the Hubble rate during BBN. In this case, ALP decays into photons may alter
the baryon-to-photon ratio and the number of relativistic degrees of freedom,
leading in particular to conflict with the well-measured ratio of the
abundances of deuterium and helium. These constraints were updated in ref.
Millea:2015qra  and shown to strongly disfavour the triangular region between
the bounds from beam dump experiments and the constraints from SN 1987A and HB
stars. Nevertheless, these constraints rely on the assumptions of a standard
cosmological history and can be significantly weakened for example if
additional relativistic degrees of freedom are present during BBN, making them
more model-dependent than the other constraints considered here. We will
therefore not show the BBN constraints in the following.

##  4 ALPs coupled to dark matter

In this section we will extend the model discussed above and consider the case
of ALPs coupled to DM particles. The immediate consequence of such a coupling
is that, provided the DM particle is sufficiently light, the ALP obtains an
invisible decay mode. This will enhance the sensitivity of searches based on
missing energy and suppress the sensitivity of searches that rely on the
reconstruction of a visible final state.

For concreteness let us assume that the DM particle is a Majorana fermion
$\chi$. The generic interaction with ALPs is then of the form

$\mathcal{L}_{\text{DM}}=g_{a\chi\chi}\,\bar{\chi}\gamma^{\mu}\gamma^{5}\chi\,\partial_{\mu}a\;,$
(19)

where $g_{a\chi\chi}$ has mass dimension $-1$, just like $g_{a\gamma\gamma}$.
The invisible decay width is then given by

$\Gamma_{\text{inv}}=\frac{g_{a\chi\chi}^{2}\,m_{a}\,m_{\chi}^{2}}{\pi}\sqrt{1-\frac{4\,m_{\chi}^{2}}{m_{a}^{2}}}\;,$
(20)

where $m_{\chi}$ denotes the DM mass and $\Gamma_{\text{inv}}=0$ for
$m_{\chi}>m_{a}/2$. If the coupling $g_{a\chi\chi}$ is large compared to
$g_{a\gamma\gamma}$ and $m_{\chi}$ is not much lighter than $m_{a}$, the
invisible decay width can easily become large relative to the visible
one.101010This is for example the case if we assume that $g_{a\gamma\gamma}$
is generated at loop-level, whereas $g_{a\chi\chi}$ is generated at tree-
level. It is then very plausible that $g_{a\chi\chi}$ is several orders of
magnitude larger than $g_{a\gamma\gamma}$.

Let us assume that this is indeed the case and that
$\text{BR}(a\to\chi\chi)\approx 100\%$. Relevant experimental constraints then
come from mono-photon searches at LEP and BaBar (as well as from LEP bounds on
$Z\to\gamma+\text{inv}$ if the ALP couples to hypercharge). The constraints
from SN 1987A will also become much stronger, as the DM particles produced in
ALP decays can contribute to supernova cooling even in parameter regions where
the ALP decay length and mean free path are small.111111 In principle even DM
particles could be trapped inside the supernova, if their interaction rate
with real photons ($\chi\gamma\to\chi\gamma$) and with virtual photons ($\chi
p\to\chi p+\gamma$) is large enough. The rate of these processes is however
suppressed relative to the rate of ALP back-conversion by a factor of
$g_{a\chi\chi}^{2}\,m_{\chi}^{2}\ll 1$. We find that the DM mean free path is
large compared to the size of the supernova as long as
$m_{\chi}\,g_{a\gamma\gamma}\,g_{a\chi\chi}\lesssim
10^{-7}\,\mathrm{GeV^{-1}}$.

Before we compare the sensitivity of the different searches, let us briefly
discuss a related question: Is it possible within the simple model introduced
above to reproduce the observed DM relic abundance while at the same time
evading constraints from various DM searches? If the process
$\chi\chi\to\gamma\gamma$ has a cross section close to the thermal one,
$(\sigma v)_{\text{th}}\approx 3\times 10^{-26}\mathrm{cm^{3}/s}$, the DM
particle can in principle obtain its relic abundance from thermal freeze-out.
For $m_{\chi}\ll m_{a}$ the annihilation cross section is given by

$\sigma(\chi\chi\to\gamma\gamma)v\simeq\frac{4\,g_{a\gamma\gamma}^{2}\,g_{a\chi\chi}^{2}\,m_{\chi}^{6}}{\pi\,m_{a}^{4}}\;,$
(21)

where $v$ denotes the relative velocity of the two DM particles in the CM
frame. This cross section is however tiny even for optimistic parameter
choices. For example, choosing $g_{a\gamma\gamma}=10^{-3}\>\text{GeV}^{-1}$,
$g_{a\chi\chi}=10^{-2}\>\text{GeV}^{-1}$, $m_{\chi}=1\>\text{GeV}$ and
$m_{a}=5\>\text{GeV}$, one obtains $\sigma v\approx 2\times
10^{-30}\mathrm{cm^{3}/s}$.

A second difficulty arises because the annihilation cross section is $s$-wave
(velocity unsuppressed). Consequently, if $\sigma v=(\sigma v)_{\text{th}}$,
the annihilation of DM into mono-energetic photons should still be observable
in the present Universe. However, searches for $\gamma$-ray lines from Fermi-
LAT exclude the thermal cross section for $s$-wave annihilation into photons
across the entire energy range Ackermann:2015lka .121212While the Fermi-LAT
collaboration only considers DM masses down to $200\,\text{MeV}$, independent
studies have extended these bounds down to $100\>\text{MeV}$ Albert:2014hwa .
We note in passing that constraints from direct detection experiments are
strongly suppressed for this scenario Frandsen:2012db  and can be neglected.
Thus, thermal freeze-out into photons is only viable if the annihilation cross
section has a strong velocity dependence.

A simple way to both enhance the annihilation cross section and to introduce a
strong velocity dependence is to consider the case that the DM mass is close
to resonance $m_{\chi}\approx m_{a}/2$. In this case, one needs to consider
the full expression for the annihilation cross section:

$\sigma(\chi\chi\to\gamma\gamma)v=\frac{g_{a\gamma\gamma}^{2}\,g_{a\chi\chi}^{2}}{4\pi}\frac{m_{\chi}^{2}\,s^{2}}{(m_{a}^{2}-s)^{2}+m_{a}^{2}\,\Gamma_{a}^{2}}\;,$
(22)

where $\sqrt{s}$ denotes the CM energy. The quantity relevant for the
calculation of the relic abundance is then the thermally averaged annihilation
cross section Gondolo:1990dk

$\langle\sigma
v\rangle=\int_{4\,m_{\chi}^{2}}^{\infty}\frac{s\sqrt{s-4\,m_{\chi}^{2}}K_{1}(\sqrt{s}/T)\sigma
v}{16\,T\,m_{\chi}^{4}\,K_{2}(m_{\chi}/T)^{2}}\mathrm{d}s\;,$ (23)

where $T$ denotes the temperature of the thermal bath and $K_{i}$ denote the
spherical Bessel functions of the second kind.

If $m_{\chi}$ is close to (but slightly below) $m_{a}/2$, the integrand in eq.
(23) is strongly peaked around $s\approx m_{a}^{2}$, such that the
intermediate ALP is on-shell and the annihilation receives a resonant
enhancement. We can then make the replacement $s\to m_{a}^{2}$ everywhere
except in the denominator of eq. (22), in which case the integration can be
performed analytically. If we furthermore substitute $\Gamma_{\text{inv}}$
from eq. (20) for $\Gamma_{a}$, we obtain the very simple expression

$\langle\sigma
v\rangle\simeq\frac{\pi\,g_{a\gamma\gamma}^{2}\,x\,K_{1}(x/r)}{64\,r^{5}\,K_{2}(x)^{2}}\;,$
(24)

where we have introduced the dimensionless temperature ratio $x=m_{\chi}/T$
and the dimensionless mass ratio $r=m_{\chi}/m_{a}$. We make two important
observations

  1. 1.

Contrary to naive expectation, $\langle\sigma v\rangle$ does not depend on
$g_{a\chi\chi}$. The reason is that increasing $g_{a\chi\chi}$ enhances the
production cross section for the ALP in the intermediate state, but at the
same time broadens the resonance. These two effects cancel exactly, so that
only the coupling to photons enters in the final expression.

  2. 2.

Even though $\langle\sigma v\rangle$ is exponentially sensitive to the mass
ratio $r$ (since $K_{1}(x/r)\propto\exp(-x/r)\sqrt{r}$), it becomes
immediately clear that it is not necessary to tune $m_{\chi}$ exactly to
$m_{a}/2$. For example, the difference between $r=0.49$ and $r=0.499$ is only
about a factor of 2 (assuming $x=20$, which is a typical value of the
temperature ratio during freeze-out).

This latter point is illustrated in figure 3, which shows in blue (dotted) the
annihilation cross section without thermal averaging, in orange (solid) the
annihilation cross section with thermal averaging and in red (dashed) the
approximate expression given in eq. (24).

Figure 3:  Comparison of the thermally averaged annihilation cross section
$\langle\sigma v\rangle$ (orange, solid) and the annihilation cross section in
the limit $v\to 0$ (blue, dotted) as a function of the mass ratio
$r=m_{\chi}/m_{a}$ for $m_{a}=1\>\mathrm{GeV}$ and
$g_{a\gamma\gamma}=10^{-4}\>\mathrm{GeV}^{-1}$. The red dashed line indicates
the simplified expression from eq. (24), which is valid close to the
resonance. Away from the resonance the cross section depends also on the ALP-
DM coupling, which has been set to $g_{a\chi\chi}=10^{-3}\>\mathrm{GeV}^{-1}$.

As we have seen above, the thermally averaged annihilation cross section (and
hence the DM relic abundance) depends only on $g_{a\gamma\gamma}$ and the mass
ratio $r$. If we fix the mass ratio $r$ we can therefore determine the value
of $g_{a\gamma\gamma}$ that yields the observed relic abundance.131313When
calculating the relic abundance by solving the Boltzmann equation, a weak
additional dependence on the DM mass arises from the fact that the number of
relativistic degrees of freedom and hence the expansion rate of the Universe
depends slightly on the temperature during freeze-out. The value of
$g_{a\gamma\gamma}$ implied by the observed relic abundance therefore depends
slightly on $m_{a}$ even for fixed mass ratio $r$. We take this dependence
into account by solving the Boltzmann equation numerically using micrOmegas
v4.2.5 Belanger:2014vza . We note that this calculation assumes that DM
couples sufficiently strongly to photons that it remains in kinetic
equilibrium during freeze-out Binder:2017rgn . It turns out that the resulting
values of $g_{a\gamma\gamma}$ are in an experimentally interesting region: For
$r=0.45$ ($r=0.49$) we find roughly $g_{a\gamma\gamma}\sim
10^{-4}\>\mathrm{GeV^{-1}}$ ($g_{a\gamma\gamma}\sim
10^{-5}\>\mathrm{GeV^{-1}}$). These estimates indicate the regions of
parameter space that are interesting for resonant thermal freeze-out.141414If
$r$ is extremely close to 0.5, the ALP-photon coupling can even be somewhat
smaller than $g_{a\gamma\gamma}=10^{-5}\>\mathrm{GeV^{-1}}$, but in this case
indirect detection constraints again become relevant, as resonant enhancement
of annihilation processes can also occur in the present Universe (see figure
3). In the following section we will compare these values to the sensitivity
that can be achieved by Belle II (see figure 4).

##  5 Sensitivity to ALPs of Belle II

The Belle II experiment at the SuperKEKB accelerator is a second generation
B-factory and successor of the Belle and BaBar experiments Abe:2010gxa . It is
currently under construction and will start data taking in 2018. SuperKEKB is
a circular asymmetric $e^{+}e^{-}$ collider with a nominal collision energy of
$\sqrt{s}$ = 10.58 GeV. The design instantaneous luminosity is $8\times
10^{35}$ cm-2 s-1, which is about 40 times higher than at the predecessor
collider KEKB. The Belle II detector is a large-solid-angle magnetic
spectrometer. Three sub-detectors are particularly relevant for the ALP
searches described in this paper: A 56-layer central drift chamber (CDC) is
used for tracking of charged particles and covers a polar angle region of
$(17-150)^{\circ}$. The electromagnetic calorimeter (ECL) comprising CsI(Tl)
crystals with an upgraded waveform sampling readout for beam background
suppression covers a polar angle region of $(12-155)^{\circ}$ and is located
inside a superconducting solenoid coil that provides a 1.5 T magnetic field.
The ECL has inefficient gaps between the endcaps and the barrel for polar
angles between $(31.3-32.2)^{\circ}$ and $(128.7-130.7)^{\circ}$. An iron
flux-return is located outside of the magnet coil and is instrumented with
resistive plate chambers and plastic scintillators to mainly detect
$K^{0}_{L}$ mesons, neutrons, and muons (KLM) that covers a polar angle region
of $(25-155)^{\circ}$.

We study the sensitivity for a dataset corresponding to an integrated
luminosity of 20 fb-1, which is expected to be collected in 2018 without
vertex detectors installed. We then scale the expected sensitivity
$S(g_{a\gamma\gamma})$ to the planned full integrated luminosity of 50 ab-1
after about 7 years of running, using
$S(g_{a\gamma\gamma})\propto\sqrt[4]{\mathcal{L}}$. We argue that the expected
increase of beam induced background rates at highest luminosity are not
relevant for these searches.

In the following we consider ALP decays into DM and into two photons from ALPs
produced in ALP-strahlung ($a\to\chi\chi$ and $a\to\gamma\gamma$) and photon-
fusion production ($a\to\gamma\gamma$ only).

###  5.1 ALP decays into dark matter

We study decays of ALPs into DM from ALP-strahlung production for ALP masses
up to $m_{a}=8.5$ GeV. Signal Monte Carlo events have been generated using
MadGraph5 v2.2.2 Alwall:2014hca . We have generated samples using a fixed ALP
mass per sample in steps of 0.05 GeV with 10,000 events each, using a
branching ratio into DM of $\text{BR}(a\to\chi\chi)=1.0$. The final state
consists of a single, highly energetic photon with an energy

$E_{\gamma}=\frac{s-m_{a}^{2}}{2\sqrt{s}}\;,$ (25)

where $\sqrt{s}=10.58$ GeV is the collision energy. This search is very
similar to the search of Dark Photon decays into DM described in ref.
lit:b2tip . The backgrounds for this search have been found to be due to high
cross section QED processes $e^{+}e^{-}\to e^{+}e^{-}\gamma(\gamma)$ and
$e^{+}e^{-}\to\gamma\gamma(\gamma)$ where all but one photon are undetected.
The background composition is a complicated function of detector geometry
details that cannot be adequately reproduced without a full Belle II detector
simulation. We therefore take the background rates from ref. lit:b2tip . It
should be noted that the irreducible background from
$e^{+}e^{-}\to\nu\bar{\nu}\gamma$ is negligible. We obtain the signal
efficiency for ALPs using generator-level Monte Carlo simulations.

Figure 4:  Present and future constraints on ALPs decaying into DM compared to
the parameter region where one can reproduce the observed DM relic abundance
via resonant annihilation of DM into photons. Note that this process is
efficient only if $m_{\chi}$ is slightly smaller than $m_{a}/2$ (see figure
3).

We determine the expected 90 % CL upper limit of signal events $n_{s}$ such
that the Poisson probability of observing less than $n$ events when expecting
$n_{s}+n_{b}$ events is $\leq 0.1$, where $n$ is the integer closest to the
number of background events $n_{b}$. Expected upper limits on the coupling
$g_{a\gamma\gamma}$ are summarized as a function of ALP mass $m_{a}$ in figure
4. The much better expected sensitivity compared to BaBar is mainly due to the
more homogeneous calorimeter of Belle II. Figure 4 also shows the parameter
ranges corresponding to resonant freeze-out. We observe that, if DM
annihilation into photons is resonantly enhanced, existing experiments are not
yet sensitive to the values of $g_{a\gamma\gamma}$ implied by the observed DM
relic abundance, but Belle II has a unique potential to probe the parameter
regions of particular interest.

The sensitivity to high mass ALPs is limited by the trigger threshold for a
single photon that will be implemented in Belle II. We conservatively assume a
trigger energy threshold of 1.8 GeV which limits the search to ALP masses
below $m_{a}$=8.6 GeV. If the trigger threshold can be lowered to
$1.2\,\text{GeV}$, the sensitivity extends to ALP masses up to $m_{a}$=9.3
GeV. A higher collision energy close to the $\Upsilon(6S)$ resonance could
further extend the sensitivity to about $m_{a}$=9.7 GeV for a trigger
threshold of 1.2 GeV.

It should be noted that while the dominant physics background for this study
comes from $e^{+}e^{-}\to\gamma\gamma(\gamma)$ events, the largest fraction of
the trigger rate for trigger thresholds $\lesssim 1.8$ GeV is due to radiative
Bhabha events $e^{+}e^{-}\to e^{+}e^{-}\gamma(\gamma)$ where both tracks are
out of the detector acceptance.

###  5.2 ALP decays into two photons

Figure 5:  Illustration of the different kinematic regimes relevant for ALP
decays into two photons with Belle II.

The experimental signature of the decays into two photons is determined by the
relation between mass and coupling of the ALP. This relation affects both the
decay length of the ALP and the opening angle of the decay photons. It leads
to four different experimental signatures (see figure 5):

  1. 1.

ALPs with a mass of $\mathcal{O(\text{GeV})}$ decay promptly, and the opening
angle of the decay photons is large enough that both decay photons can be
resolved in the Belle II detector (_resolved_).

  2. 2.

For lighter ALP masses but large couplings $g_{a\gamma\gamma}$, the decay is
prompt but the ALP is highly boosted and the decay photons merge into one
reconstructed cluster in Belle II calorimeter if $m_{a}\lesssim 150$ MeV
(_merged_).151515This corresponds to an average opening angle of about
$(3-5)^{\circ}$ in the lab system that depends on the position in the
detector.

  3. 3.

Even lighter ALPs decay displaced from the interaction point but still inside
the Belle II detector. This is a challenging signature that consists of two
reconstructed clusters, one of which has a _displaced_ vertex and contains two
merged photons. The latter two conditions typically yield a bad quality of the
reconstructed photon candidate which is not included in resolved searches with
final state photons. There is however enough detector activity in the ECL or
KLM that these are vetoed in searches for invisible final states to reduce
high rate $e^{+}e^{-}\rightarrow\gamma\gamma$ backgrounds.

  4. 4.

The lifetime of light ALPs with small couplings is large enough that a
significant fraction of ALPs decays outside of Belle II. The experimental
signature is a single photon final state that looks identical to the ALP decay
into DM (_invisible_).

The detailed Belle II sensitivity for _merged_ and _displaced_ decays depends
strongly on the actual performance of the Belle II reconstruction software and
beam background levels which are beyond the scope of this paper. It may be
possible to search for displaced clusters in the KLM due to its longitudinal
resolution of $\mathcal{O}$(10 cm). The resolved region can potentially be
extended towards smaller ALP masses if the ECL reconstruction is improved.

We study _resolved_ ALP decays into two photons
($e^{+}e^{-}\to\gamma(a\to\gamma\gamma)$) over a range of ALP masses between
$m_{a}$ = (0.05–9.0) GeV. Signal Monte Carlo events have been generated using
MadGraph5 v2.2.2 Alwall:2014hca . We have generated samples using a fixed ALP
mass per sample in steps of 0.05 GeV with 10,000 events each using a branching
ratio into photons of $\text{BR}(a\to\gamma\gamma)=1.0$.

We use a simplified geometry description of the Belle II detector to take into
account the ECL geometry acceptance. We use the ECL energy resolution
lit:b2tip  that is expected for 10 % of the full luminosity backgrounds. It
should be noted that all three photons have rather high energy and the photon
energy resolution is expected to not change significantly for even higher beam
backgrounds. We use the approximate ECL crystal positions and sizes to
estimate the performance in resolving overlapping photon clusters.

We find that the background is dominated by the QED process
$e^{+}e^{-}\to\gamma\gamma\gamma$ with three photons in the final state.
Background samples are generated using BABAYAGA.NLO CarloniCalame:2003yt ;
CarloniCalame:2001ny ; CarloniCalame:2000pz . Additional small backgrounds for
small ALP masses may arise from $e^{+}e^{-}\to\gamma\gamma$ with a third
photon candidate coming from beam-induced backgrounds, and from
$e^{+}e^{-}\to\gamma\gamma$ where one of the photon converts into an electron-
positron pair outside of the tracking detectors. The former will be reduced
using the very good time resolution $\mathcal{O}$(ns) of the Belle II ECL at
high photon energies Aulchenko:2017lmh . To reduce background from pair
conversion, one can use the fact that the secondary electron-positron pair
splits in the magnetic field and veto events where the polar angle difference
between the photons of the lowest invariant mass photon pair is small and the
azimuthal angle difference is rather large. We expect that the pair conversion
background is only relevant in the ECL backward region where significant
material from the CDC readout electronics is placed about 40 cm away from the
crystal front.

A further potential background arises from the SM processes
$e^{+}e^{-}\to\pi^{0}\gamma$, $e^{+}e^{-}\to\eta\gamma$ and
$e^{+}e^{-}\to\eta^{\prime}\gamma$ Czyz:2017veo . We therefore exclude
${}_{-75\,\text{MeV}}^{+50\,\text{MeV}}$ mass regions around the nominal
$\eta$ and $\eta^{\prime}$ masses (our analysis is not sensitive to
$m_{a}\approx m_{\pi}$). In the actual analysis a full study of these
backgrounds should be included. Finally, we assume that both beam backgrounds
and pair conversion backgrounds can be reduced to a negligible level using the
selections described above, without significantly affecting the signal
selection efficiency.

Our event selection requires three photons with a CM energy $E^{*}>0.25$ GeV
and a polar angle in the laboratory frame
$17^{\circ}<\theta^{\text{lab}}<150^{\circ}$. The invariant mass of the two
photons from the ALP decay will peak at the ALP mass. We perform the
sensitivity study twice, once using all three possible photon pair
combinations (high mass selection) and once using only the photon pair
combination with the lowest invariant mass (low mass selection). The latter
has a smaller signal efficiency especially at higher ALP masses but a lower
combinatorial background. For the three photon combination case we select
events where the maximum absolute cosine of the three helicity angles is less
than 0.9, and for the two photon combination case we keep events where the
absolute cosine of the helicity angle is less than 0.6. These selection
criteria maximize the ratio of $\sqrt{S}/B$, where $S$ is the number of signal
events and $B$ is the number of background events, after all other selection
criteria have been applied. It should be noted that the helicity selection
criteria not only reduce $e^{+}e^{-}\to\gamma\gamma\gamma$ backgrounds, but
will also suppress backgrounds from $e^{+}e^{-}\to\gamma\gamma$ combined with
a random third photon from beam backgrounds. We require that all three photons
are separated by at least 2 ECL crystals in both polar and azimuthal
direction. We do not constrain the three photon invariant mass to the
collision energy since our MadGraph signal Monte Carlo does not include
additional photon radiation whereas the background Monte Carlo does.

Figure 6:  Belle II $3\gamma$ efficiency as function of ALP mass after the
final selection. The different low mass selections correspond to a minimum
photon separation of 1, 2, and 4 crystals in the ECL which is an approximation
for the expected performance of an improved reconstruction, the default
reconstruction and the reconstruction in the first trigger level (see text for
details).

We finally select candidates within $[-3\sigma_{m_{2}},+1.5\sigma_{m_{2}}]$
around the generated ALP mass, where $\sigma_{m_{2}}$ is the invariant mass
resolution of the decay photon pair. For high mass ALPs we select events
within $[-3\sigma_{\gamma},+1.5\sigma_{\gamma}]$ around the expected recoil
photon energy (see equation 25) instead. The ranges contains about 85 % of the
previously selected signal events. The signal efficiency after all selections
is flat and about (35–40) % ((50–55) %) for the two photon (three photon)
combination (see figure 6). The photon angle separation distance of 2 ECL
crystals is a conservative estimate of the Belle II offline reconstruction
performance and can likely be improved using advanced reconstruction
techniques based on Machine Learning methods, and by using shower shape
techniques similar to those applied in high energy $\pi^{0}$ reconstruction.
We show the efficiency for single ECL crystal difference for comparison as
well.

Events from $e^{+}e^{-}\to\gamma(a\to\gamma\gamma)$ are typically triggered by
three energy depositions of at least 0.1 GeV in the ECL. Unlike in the Belle
II offline reconstruction, the photon reconstruction at trigger level is much
simpler and has a worse angular separation power. We expect that a separation
of less than 4 ECL crystals will result in merged photon clusters and make
this trigger inefficient for ALP masses below about 0.5 GeV. An ideal trigger
will require at least two highly energetic ECL clusters and must not satisfy
$e^{+}e^{-}\to e^{+}e^{-}$ (Bhabha) vetoes. However, any
$e^{+}e^{-}\to\gamma\gamma$ veto decision must be delayed to the high level
trigger where offline reconstruction is available in order to maintain a high
trigger efficiency for low mass ALPs.

Figure 7:  Projected Belle II sensitivity (90 % CL) compared to existing
constraints on ALPs with photon coupling (left) and hypercharge coupling
(right), as well as the projected sensitivities from SHiP Dobrich:2015jyk  and
the LHC Bauer:2017ris .

We obtain the expected 90 % CL sensitivity as described above. The sensitivity
for long-lived ALPs decaying into two photons is determined from the
sensitivity of ALP decays into DM, taking into account the reduced efficiency
given by eq. (6) using a detector length161616The event selection includes a
veto of energy depositions in the KLM. The detector length is hence taken as
approximate outer radius of the barrel KLM. of $L_{\mathrm{D}}=300$ cm
Abashian:2000cg . The projected sensitivities to the coupling
$g_{a\gamma\gamma}$ are summarized as a function of ALP mass $m_{a}$ in figure
7.

We make a number of important observations from figure 7. First of all, we
note that for very light ALPs (i.e. $m_{a}\sim 1\,\mathrm{MeV}$) Belle II
single-photon searches can push significantly beyond current constraints from
beam dump experiments and can potentially explore the triangular region around
$g_{a\gamma\gamma}\sim 10^{-5}\,\mathrm{GeV^{-1}}$, which is currently only
constrained by model-dependent cosmological considerations. For heavier ALPs
(i.e. $150\,\mathrm{MeV}<m_{a}<10\,\mathrm{GeV}$) Belle II searches for three
resolved photons can significantly improve over existing bounds from LEP even
with early data ($20\,\mathrm{fb^{-1}}$). With larger data sets, Belle II will
be able to probe couplings of the order of $g_{a\gamma\gamma}\sim
10^{-4}\,\mathrm{GeV^{-1}}$ over a wide range of ALP masses. Improvements in
the Belle II reconstruction software could push the sensitivity for three
resolved photons to slightly lower masses and a dedicated search for displaced
photons could extend the long-lived search towards higher masses.

Comparing the two panels in figure 7 we note furthermore that there is a
remarkable complementarity between Belle II, SHiP and LHC. SHiP will have
greatest sensitivity in the parameter region where the ALP decay length is
$\mathcal{O}(1\text{--}100)\,\mathrm{m}$, which is difficult to explore with
Belle II and the LHC. The LHC, on the other hand, is sensitive mostly to the
coupling $g_{a\gamma Z}$, while Belle II and SHiP directly probe the ALP-
photon coupling $g_{a\gamma\gamma}$. The combination of these experiments will
therefore allow to make significant progress in the exploration of the ALP
parameter space. Moreover, we can hope to see an ALP signal in more than one
experiment, which would potentially enable us to reconstruct its properties
and coupling structure.

###  5.3 Photon fusion

So far we have focused on the case that the ALP is produced in association
with a highly-energetic photon, which facilitates an efficient reconstruction
of these events. For ALPs produced in photon fusion the situation becomes more
complicated, as the transverse momenta of electron and positron after the
collision (and hence their polar angle) are too small to be detectable.

Figure 8:  Comparison of ALP production in $e^{+}e^{-}$ collisions via ALP-
strahlung and via photon fusion. The left panel shows the total cross section,
the right panel the differential cross section with respect to the
longitudinal momentum $p_{z}$.

Searches for ALPs produced in photon fusion are interesting for two reasons:
First, as shown in the left panel of figure 8 the total ALP production cross
section from photon fusion significantly exceeds the one from ALP-strahlung
(in particular for small ALP masses), so that photon fusion is responsible for
the vast majority of ALPs produced at Belle II Marciano:2016yhf ;
Izaguirre:2016dfi . And second, the production cross section from photon
fusion peaks for small ALP momenta, i.e. ALPs will be produced dominantly at
rest (see right panel of figure 8). This means that, in contrast to ALP-
strahlung, the opening angle between the two photons produced in the ALP decay
will typically be large even for low-mass ALPs.

The signature in the Belle II detector will consist of two photons with an
invariant mass equal to the ALP mass and missing energy along the beam-pipe.
The azimuthal angles of the two photons are back-to-back in the CM frame. The
Belle II acceptance for ALPs produced in photon fusion is high: For
$m_{a}=0.2$ GeV ($m_{a}=2.0$ GeV) 66 % (89 %) of all ALPs have both decay
photons in the ECL acceptance. However, for low mass ALPs the photon energy is
small and often below a typical trigger threshold of 100 MeV per ECL cluster.
Studies have shown a very large beam-induced background of low energy ECL
clusters lit:b2tip , making the detection of ALPs produced in photon fusion
very difficult. While it may be possible to reject a part of these events in
the offline reconstruction, the trigger rates are probably too large to
sustain.

A possible opportunity might be to consider the case where either the electron
or the positron receives sufficiently large transverse momentum to be tagged
Izaguirre:2016dfi . However, the cross section is strongly reduced in this
case and backgrounds remain rather large compared to the $3\gamma$ final
state. A detailed study of the potential sensitivity of Belle II in this
search channel is beyond the scope of this work.

##  6 Conclusions

In this work we have discussed the experimental situation concerning axion-
like particles (ALPs) that interact with Standard Model particles dominantly
via couplings to photons and electroweak gauge bosons. Reviewing existing
constraints we have argued that the bounds from $e^{+}$-$e^{-}$ colliders
conventionally shown for this scenario are outdated. We have updated these
constraints by reinterpreting mono-photon searches at LEP and Dark Photon
searches at BaBar. We have furthermore investigated the bounds on ALPs from
electron beam dumps and from SN 1987A and provided refined estimates of these
constraints. We have also discussed the case that ALPs can be produced in the
decay $Z\to a\gamma$, which is relevant if ALPs couple to hypercharge. A
summary of existing constraints is shown in figure 2.

A scenario of particular interest is ALPs coupled to a light DM particle,
which induces invisible ALP decays and hence removes a number of experimental
constraints. In this case DM can pair-annihilate into photons via ALP
exchange. We pointed out that if these annihilations are resonantly enhanced
in the Early Universe, the DM particle can be a thermal relic with the
required abundance and satisfy all experimental and observational constraints.
The DM relic abundance then depends on only two parameters (the ALP-photon
coupling and the ratio of DM mass to ALP mass), leading to a highly predictive
scenario. Depending on how close the mass ratio is to resonance ($m_{\chi}\sim
m_{a}/2$), the observed DM relic abundance implies ALP-photon couplings in the
range $g_{a\gamma\gamma}\sim 10^{-4}\text{--}10^{-5}\,\mathrm{GeV}^{-1}$.
These values are consistent with all existing experimental constraints but lie
precisely in the parameter region that can be probed with single-photon
searches at Belle II.

Our central observation is that existing bounds on ALPs can be significantly
improved with single-photon searches and searches for three resolved photons
at Belle II. The former type of search probes invisibly decaying ALPs as well
as ALPs that escape from the detector before decaying, while the latter search
strategy is highly sensitive to ALPs in the GeV region, which decay promptly
into a pair of photons. We have shown that the sensitivity of single-photon
searches is substantially better than has been estimated previously due to a
significant reduction in background compared to BaBar. In combination these
search strategies can cover wide ranges of ALP parameter space and explore
various models that are of interest for both cosmology and particle physics
(see figures 4 and 7). We have found searches with Belle II to be highly
complementary to searches with SHiP (which probes different ALP decay lengths)
and at the LHC (which probes different coupling structures).

We have pointed out that there are two regimes that require additional studies
with full Belle II simulations in order to understand the sensitivity: ALP
decays from displaced vertices and prompt decays of highly-boosted ALPs
leading to a merging of the two photons. Finding ways to extend the Belle II
sensitivity in these regions will be of great interest for future work.
Searching for ALPs produced via photon fusion is one potential avenue to make
progress, but backgrounds for this search channel may be prohibitive. Future
progress in the search for ALPs therefore requires both the implementation of
the search strategies described in this work as well as the exploration of
innovative approaches that can reach the remaining corners of parameter space.
Combining these with a better understanding of astrophysical constraints of
ALPs may finally enable us to understand the role that ALPs play in the
Universe.

###### Acknowledgements.

We would like to thank Babette Döbrich and Joerg Jaeckel for helpful comments
on the manuscript and Rebecca Leane, Josef Pradler, Georg Raffelt, Diego
Redigolo, Javier Redondo, Brian Shuve and Martin W. Winkler for useful
discussions. This work is supported by the German Science Foundation (DFG)
under the Collaborative Research Center (SFB) 676 “Particles, Strings and the
Early Universe” and the Emmy Noether Grant No. KA 4662/1-1, the ERC Starting
Grant ‘NewAve’ (638528) as well as NSERC (Canada).

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