the zero background hypothesis is assumed. Studies performed with the already
acquired $3\times 10^{16}$ pot dataset in dump mode show that the background
can be reduced to zero with the current setup for fully reconstructed final
states, while for open final states the addition of an Upstream Veto in front
of the decay volume is required. The addition of this detector is currently
under study by the Collaboration.

Figure 30: BC6: Sensitivity to Heavy Neutral Leptons with coupling to the
first lepton generation only. Current bounds (filled areas) and near ($\sim$ 5
years) future physics reach of two PBC projects, FASER and NA62++ (solid
lines). See text for details.

Physics reach of PBC projects on 10-15 year timescale

On 10-15 year timescale many PBC projects can contribute to this benchmark
case, as shown in Figure 31: MATHUSLA200, FASER2, CODEX-b and SHiP. For
MATHUSLA200 we show separately the contributions from heavy mesons and gauge
bosons decays. The SHiP sensitivity curve is obtained without (solid curve)
and with (dashed curve) a particular assumption for the contribution from
$B_{c}$. This is because the $\sigma(B_{c})/\sigma(B)$ fraction at the SPS
energies is not reliably known. We therefore show an upper sensitivity limit
provided by assuming the fraction measured at the LHC energy.

Also in this case the plot shows the 90% CL exclusion limits under the
hypothesis of zero background. This hypothesis is a strong assumption that has
been properly validated only by SHiP, so far, using a full GEANT4 simulation
of the detector, including digitization and reconstruction, and large samples
of Monte Carlo data. The background evaluation for MATHUSLA200, CODEX-b and
FASER2 is still work in progress and will be carried on in the coming years.
Figure 31 shows also projections from the LBNE near detector as 5-year
sensitivity corresponding to an exposure of $5\times 10^{21}$ protons on
target for a detector length of 30 m and assuming a normal hierarchy of
neutrinos masses Adams:2013qkq  and from FCC-ee with $10^{12}$ $Z^{0}$ decays
and HNLs decaying between 10-100 cm from the interaction vertex
Blondel:2014bra .

Figure 31: BC6: Sensitivity to Heavy Neutral Leptons with coupling to the
first lepton generation only. Current bounds (filled areas) and 10-15 years
prospects for PBC projects (SHiP, MATHUSLA200, CODEX-b and FASER2) (solid
lines). Projections for a LBNE near detector with $5\times 10^{21}$ pot and
from FCC-ee with $10^{12}$ $Z^{0}$ decays are also shown.

####  9.3.2 Neutrino portal with muon-flavor dominance (BC7)

In this Section we consider the case in which HNLs couple only to second SM
generation and the sensitivity plots are shown in the plane
{$|U_{\mu}|^{2},m_{N}$}.

Current bounds, experimental landscape and PBC projects on 5 year timescale

Current bounds and the future experimental landscape in the next $\sim$ 5
years, including some PBC projects, are shown in Figure 32 for the case of HNL
with couplings only to the second lepton generation and masses in the MeV-GeV
range.

Also in this case the allowed range of couplings is bounded from below by the
BBN constraint Ruchayskiy:2012si , and the see-saw limit Canetti:2010aw .
Existing experimental limits are shown as filled coloured areas: for masses
below the charm mass they arise mostly from the same beam dump experiments
contributing to the sensitivity for electron-flavor dominance (PS191
Bernardi:1987ek  and CHARM Bergsma:1985is , as explained in the previous
paragraph) with the addition of NuTeV Vaitaitis:1999wq , and E949
Artamonov:2014urb :

  * -

NuTeV @ Fermilab: a search for HNLs decaying in muonic final states has been
performed at the neutrino detector NuTeV at Fermilab in 1996-1997, using
$2\times 10^{18}$ 800 GeV protons interacting with a Berillium-oxide target
and a proton dump. Ref. Vaitaitis:1999wq .

  * -

E949 @ BNL: Evidence of a heavy neutrino, in the process
$K^{+}\to\mu^{+}\nu_{R}$ was sought by the E949 collaboration using $1.7\times
10^{12}$ stopped kaons.

Above the charm mass, current bounds are set by DELPHI Abreu:1996pa , Belle
Liventsev:2013zz , CMS Sirunyan:2018mtv  with the same analysis used to set
bounds for electron-dominance, and by LHCb with a dedicated analysis to search
for prompt and diplaced $\pi^{-}\mu^{+}$ vertices in
$B^{+}\to\pi^{-}\mu^{+}\mu^{+}$ LNV decays Aaij:2014aba .

As in the benchmark case of HNLs with couplings only to the first generation,
NA62++ in dump mode and FASER will be able to perform this search with
competitive physics reach in a time scale of $\sim$ 5 years.

Figure 32: BC7: Sensitivity to Heavy Neutral Leptons with coupling to the
second lepton generation only. Current bounds (filled areas) and near ($\sim$
5 years) physics reach of two PBC projects, FASER and NA62++ (solid lines).
See text for details.

Physics reach of PBC projects on 10-15 year timescale

Figure 33 shows the 90 % CL exclusion limits from MATHUSLA200, FASER2, CODEX-b
and SHiP in a 10-15 years time scale. Also in this case the curves are
obtained under the assumption of zero background, for which the same
considerations drawn in the previous paragraph hold.

Figure 33: BC7: Sensitivity to Heavy Neutral Leptons with coupling to the
second lepton generation only. Current bounds (filled areas) and 10-15 years
prospects for PBC projects (SHiP, MATHUSLA200, CODEX-b and FASER2) (dotted and
solid lines). Projections for the LBNE near detector with $5\times 10^{21}$
pot and FCC-ee with $10^{12}$ $Z^{0}$ decays are also shown.

####  9.3.3 Neutrino portal with tau-flavor dominance (BC8)

In this Section we consider the case in which HNLs couple only to third SM
generation and the sensitivity plots are shown in the plane
{$|U_{\tau}|^{2},m_{N}$}.

Current bounds and experimental landscape

Current bounds and future experimental landscape in the next $\sim$ 5 years,
including some PBC projects, is shown in Figure 34 for the case of HNL
coupling only to the third lepton generation and masses in the MeV-GeV range.
Also in this case the allowed range of couplings is bounded from below by the
BBN constraints Ruchayskiy:2012si , and the see-saw limit Canetti:2010aw .

Main bounds in this benchmark case arise from CHARM Orloff:2002de , NOMAD
Astier:2001ck , and again the same data from DELPHI Abreu:1996pa  used for the
other two benchmark cases (BC6 and BC7).

  * -

CHARM: limits on the square mixing strength $|U_{\tau}|^{2}$ in a mass range
10-290 MeV were set by re-interpreting the null result of a search for events
produced by the decay of neutral particles into two electrons performed by the
CHARM experiment using the neutrino flux produced by ${\mathcal{O}}(2\times
10^{18})$ 400 GeV protons on a solid copper target. Ref. Orloff:2002de .

  * -

NOMAD: a search for heavy neutrinos was performed using $4.1\times 10^{19}$
450 GeV protons on target at the WANF facility at CERN in 1996-1998. The HNLs
were searched in the process $D_{s}\to\tau\nu_{R}$ followed by the decay
$\nu_{R}\to\nu_{\tau}e^{+}e^{-}$ in the NOMAD detector. This allowed to derive
an upper limit on the mixing strength between the heavy neutrino and the tau
neutrino in the $\nu_{R}$ mass range from 10 to 190 MeV. Ref. Astier:2001ck .

Physics reach of PBC projects on 5 and 10-15 year timescale

Among the PBC projects the only two contributing on a 5-year timescale are,
again, FASER with 150 fb-1 and NA62++, as shown in Figure 34. Figure 35 shows
the 90 % CL exclusion limits from MATHUSLA200, FASER2, CODEX-b and SHiP in a
10-15 years time scale. The physics reach from FCC(ee) with $10^{12}$ $Z^{0}$
is also shown.

Figure 34: BC8: Sensitivity to Heavy Neutral Leptons with coupling to the
third lepton generation only. Current bounds (filled areas) and near ($\sim$ 5
years) future physics reach of two PBC projects, FASER and NA62++ (solid
curves). See text for details.

Figure 35: BC8: Sensitivity to Heavy Neutral Leptons with coupling to the
third lepton generation only. Current bounds (filled areas) and 10-15 years
prospects for PBC projects (SHiP, MATHUSLA200, CODEX-b and FASER2) (solid and
dotted curves). Projections from FCC-ee with $10^{12}$ $Z^{0}$ decays are also
shown.

###  9.4 Axion Portal

The discovery of the Higgs boson shows clearly that elementary scalar bosons
exist in nature. Therefore it is timely and well-motivated to search for
further light scalar or pseudoscalar particles. Pseudo-scalar particles can
arise as pseudo-Nambu-Goldstone bosons (PNGB) of a spontaneously broken _U(1)_
symmetry at a scale $f_{A}$. The principal example of very light pseudo-
Goldstone bosons is the axion Peccei:1977hh ; Peccei:1977ur ; Wilczek:1977pj ;
Weinberg:1977ma  introduced to solve the strong _C_ P problem in QCD. Natural
extensions of the axion paradigm bring to a wide range of interesting
pseudoscalar particles which typically have very similar interactions as the
axion, but without a strict relation between the mass and the coupling, the
Axion-Like Particles or ALPs.

ALPs also provide an interesting connection to the puzzle of dark matter,
because they can mediate the interactions between the DM particle and SM
states and allow for additional annihilation channels relevant for the thermal
freeze-out of DM. In fact in presence of an additional pseudoscalar particle
that mediates the interactions of DM with the SM sector Freytsis:2010ne ;
Dienes:2013xya , constraints from direct detection experiments Akerib:2013tjd
; Agnese:2013jaa ; Agnese:2014aze ; Angloher:2014myn  and invisible Higgs
width Aad:2014iia ; Chatrchyan:2014tja  on the scalar portal can be easily
evaded Freytsis:2010ne ; Dienes:2013xya .

Moreover, if the pseudoscalar mass is in the sub-GeV range it can furthermore
evade detection at the LHC, as, e.g., in monojet searches Chatrchyan:2011nd .
Another advantage of a very light pseudoscalar $a$ is that it allows for the
possibility that DM can obtain the correct relic density from thermal freeze-
out even if it is very weakly coupled to SM particles. This is due to the fact
that, provided the pseudo-scalar mass is less than twice the mass of the DM
particle $\chi$, the annihilation process $\chi\overline{\chi}\to aa$,
followed by decays of the pseudoscalars into SM particles, allows for a highly
efficient annihilation of DM particles. The only important constraint is that
such pseudoscalar particles must decay before BBN. As explained in Section
2.1, ALPs can mediate interactions between DM and the SM sector via three
different couplings, photon-, gluon-, and fermion-coupling.

####  9.4.1 Axion portal with photon-coupling (BC9)

Assuming a single ALP state $a$, and the predominant coupling to photons, all
phenomenology (production, decay, oscillation in the magnetic field) can be
determined as functions on ($m_{a};g_{a\gamma\gamma}=f_{\gamma}^{-1}$)
parameter space.

Current bounds and near future experimental landscape

The current bounds for ALPs with photon coupling are shown in Figure 36, left.
A zoom on the region of interest for experiments at accelerators is shown in
the right panel and covers a range of masses between MeV and GeV. We note that
this is also the mass region of interest in models where ALPs serve as
mediators to a DM sector.

Searches for ALPs with photon coupling have been performed using monophoton
searches at LEP, $e^{+}e^{-}\to\gamma^{*}\to a\gamma$, mono-photon searches at
BaBar ($e^{+}e^{-}\to\gamma a,a\to$ invisible), radiative $\Upsilon$ decays
($\Upsilon(nS)\to\gamma^{*}\to\gamma a$), radiative $Z-$boson decays, and
electron- and proton beam dump experiments, where the ALPs are produced mainly
via the Primakoff effect, i.e. the conversion of a photon into an ALP in the
vicinity of a nucleus Halprin:1966zz .

Figure 36: Left panel: current limits on axion with photon coupling in the
plane coupling ($g_{a\gamma\gamma}$) versus mass ($m_{a}$). The right panel
shows the zoom in the range of masses interesting for accelerator-based
experiments (note the different units for the mass ranges in the two panels).
In the left pane, current bounds are shown as filled coloured areas. A
possible sensitivity from Belle-II from a phenomenological study Dolan:2017osp
is also shown.

  * -

E141 @ SLAC (electron beam dump): primarily searched for long-lived particles
decaying into the $e^{+}e^{-}$ final state Riordan:1987aw  but the addition of
a photon converter in front of the detector for a a certain period of data
taking opened the possibility to be sensitive also to photons from an ALP
decay. Reinterpretation of the E141 results was performed in Refs.
Krasny:1987eb ; Dobrich:2017gcm , leading to a somehow more conservative
bounds with respect previous interpretations Hewett:2012ns .

  * -

E137 @ SLAC (electron beam dump): dedicated search for ALPs coupling only to
photons Bjorken:1988as . However the exclusion limits presented in the paper
did not contain the turnover towards large couplings due to the exponential
suppression of the number of ALPs that reach the detector, this has been added
in Ref. Dolan:2017osp .

  * -

CHARM (proton beam dump): a search for a ALP in 400 GeV proton interactions
with a thick copper target was performed with the CHARM detector
Bergsma:1985qz . The target was placed 480 m apart from the 35 m long decay
volume and the hypothetical decay $a\to\gamma\gamma$ has been sought using a
fine-grained calorimeter of 9 m2 active area.

  * -

NuCal (proton beam dump): The production and decay of a light scalar and
pseudoscalar particles has been investigated in a proton-iron beam dump
experiment at the 70 GeV Serpukhov accelerator. Ref. Blumlein:1990ay .

  * -

LEP: Limits from LEP data in the ALPs mass range MeV - GeV arise from a
reinterpretation Jaeckel:2015jla  of the LEP $Z^{0}\to\gamma\gamma$ data
Acciarri:1994gb ; Abreu:1991rm ; Abreu:1994du ; Acciarri:1995gy  where one of
the two neutral clusters was considered the result of a merging of two highly
collimated photons from the ALP decay in the process $Z^{0}\to
a\gamma,a\to\gamma\gamma$. Mono-photon searches, i.e. searches for highly-
energetic photons in association with missing energy resulting from the
process $e^{+}e^{-}\to\gamma+a(a\to$ invisible) have been performed as well
Abdallah:2008aa  but they are not sensitive to ALPs with mass in the sub-GeV
range Dolan:2017osp .

  * -

Bound from Astrophysics: Supernova 1987A. Weakly coupled particles such as
axions or ALPs with masses up to about 100 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 Dolan:2017osp .

We note that mono-photon searches have also been carried out at the LHC (for
the most recent analyses see e.g. ref. Aaboud:2017dor , but their sensitivity
does not significantly improve on the bound from LEP.

Near future bounds will come from Belle-II where the ALP can be searched in
the invisible and 3 $\gamma$ decay modes. A phenomenological study has been
performed in Ref. Dolan:2017osp  where the authors consider ALP decays into
dark matter (invisible) and two (resolved) photons. The bounds have been shown
in Figure 36. However the $a\to$ invisible sensitivity heavily relies on the
possibility to use the single-photon trigger with a low threshold (1.8 GeV),
that in not guaranteed at the nominal Belle-II luminosity regime.

PBC projects on 5 and 10-15 years timescale

Three PBC experiments can perform searches of ALPs with photon coupling on a 5
year timescale: NA62++ in dump mode and FASER, will look for visible ALP
decays, $a\to\gamma\gamma$, while NA64${}^{++}(e)$ will be able to perform a
search into visible and invisible decays. The contour plots are shown in
Figure 37. On a longer timescale the big PBC projects can enter in the game
further extending the physics reach in a still uncharted parameter space. On
this respect SHiP and LDMX will be fully complementary, the first covering
larger masses and smaller coupling, the latter filling the uncovered phase
space between the old beam-dump experiments and the colliders, in the 10-300
MeV mass range.

Figure 37: BC9: ALPs with photon coupling. Current bounds (filled areas) and
prospects for PBC projects on 5 years timescale (solid lines) in the plane
coupling $g_{a\gamma\gamma}$ versus mass $m_{\rm ALP}$. The results from a
phenomenological study for Belle-II Dolan:2017osp  is also shown.

Figure 38: BC9: ALPs with photon coupling. Current bounds (filled areas) and
prospects for PBC projects on 10-15 years timescale (solid lines) in the plane
coupling $g_{a\gamma\gamma}$ versus mass $m_{\rm ALP}$. The results from a
phenomenological study for Belle-II Dolan:2017osp  is also shown.

####  9.4.2 Axion portal with fermion-coupling (BC10)

Assuming a single ALP state $a$, and the predominant coupling to fermions, all
phenomenology (production and decay) can be determined as functions on
($m_{a};g_{Y}=2vf^{-1}_{\ell},2vf^{-1}_{q}$), with $v$ the vev of the Higgs.
Furthermore, for the sake of simplicity, we take $f_{q}=f_{\ell}$. Details
about approximations and assumptions in computing sensitivities for this
benchmark case are reported in Appendices A and B.

The effective Yukawa coupling ALP-SM fermions is proportional to the mass of
the SM fermions. Hence: a possible ALP with fermion coupling is mostly
originated from meson decays and only very rarely from electrons. Heavy mesons
can be produced in $e^{+}e^{-}$ colliders, $pp$ colliders and in the
interactions of a proton beam with a target.

Searches for ALPs with fermion coupling are being pursued at the LHC, namely
in the analysis of rare $B$ decays as for example $B^{+}\to
K^{*0}\mu^{+}\mu^{-}$ Aaij:2015tna . The geometry of the LHC experiments, on
the other hand, is such that a search can be performed only if the ALP decays
more or less instantaneously, hence has large couplings. For ALPs with smaller
couplings and longer lifetimes these searches are much less effective even
though ALPs may still be produced in abundance.

Beam-dump experiments in contrast are particularly sensitive to long-lived and
very weakly coupled light new states, which can travel through the hadron
absorber before decaying. Several constraints already exist, mostly coming
from old beam dump experiments as CHARM Bergsma:1985qz , NuCal Blumlein:1990ay
, and E613 Duffy:1988rw . Other constraints are derived from $K$ and $B$
mesons experiments, as explained below.

Current bounds and near future prospects, including PBC projects

The current status of the exclusion limits for ALPS with fermion coupling in
the MeV-GeV range is shown in Figure 39, as filled coloured areas.

Most of the current bounds arise from a re-interpretation of experimental
results from CHARM Bergsma:1985qz , E949 Anisimovsky:2004hr ; Artamonov:2009sz
, KTeV AlaviHarati:2000hs  performed by theorists Dolan:2014ska ;
Dobrich:2018jyi . As such, these bounds should be taken with many caveats. A
few searches are instead coming directly from experiments, as for example
BaBar Lees:2012iw  and LHCb Aaij:2015tna ; Aaij:2016qsm .

  * -

$K^{+}\to\pi^{+}+X:$ the $K_{\mu}2$ experiment has studied the momentum
distribution of charge pions produced in the decay $K^{+}\to\pi^{+}$
Yamazaki:1984vg . In presence of a light pseudoscalar, the decay channel
$K^{+}\to\pi^{+}a$ would lead to a bump in the spectrum.

  * -

$K^{+}\to\pi^{+}+$ invisible: reinterpretation of the E949 results
Anisimovsky:2004hr ; Artamonov:2009sz  as an upper bound on the process
$K^{+}\to\pi^{+}a$ performed in Ref. Dolan:2014ska , and cross-checked by the
KLEVER collaboration. The curve assumes that the ALP escapes the decay volume.

  * -

$B^{0}\to K_{S}$ \+ invisible: This search is the analogous as the
$K^{+}\to\pi^{+}$ \+ invisible search. The strongest bounds come from Cleo
Ammar:2001gi .

  * -

$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$: in the mass range $210<m<420$ MeV the
pseudoscalar will decay predominantly to muon pairs. The KTeV/E749
collaboration has set an upper limit on the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$
decay AlaviHarati:2000hs  and this result has been converted into an upper
limit for the branching fraction of the decay $K_{L}\to\pi^{0}a$ under the
hypothesis that the ALP decays instantaneously Dolan:2014ska .

  * -

Radiative $\Upsilon$ decays: pseudo-scalar particles have been sought by the
BaBar collaboration in radiative $\Upsilon$ decays $\Upsilon\to a\gamma$, with
$a\to\mu^{+}\mu^{-}$ for $a<2m_{\tau}$ Lees:2012iw  and $a\to\tau^{+}\tau^{-}$
for $m_{a}$ above threshold Lees:2012te .

  * -

$B\to K\mu^{+}\mu^{-}$: the measurement of the branching fraction of the
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ decay as a function of the dimuon mass
performed by LHCb Aaij:2012vr  has been interpreted in Ref. Dolan:2014ska  as
an upper bound for the process $B^{+}\to K^{+}a,a\to\mu^{+}\mu^{-}$ in each
$\mu^{+}\mu^{-}$ mass bin, under the hypothesis that $a$ decays
instantaneously. Dedicated searches have been instead performed by the
Collaboration, allowing also for displaced dimuon vertices in the $B^{0}\to
K^{0*}\mu^{+}\mu^{-}$ and $B^{+}\to K^{+}\mu^{+}\mu^{-}$ processes, and the
results reported in Refs. Aaij:2015tna  and Aaij:2016qsm , respectively. These
data have been adapted to the model prescriptions used in this study by F.
Kahlhoefer.

Figure 39: BC10: ALPs with fermion coupling. Current bounds (filled areas) and
near ($\sim$ 5 years) prospects for PBC projects (solid lines). CHARM and LHCb
filled areas have been adapted to PBC prescriptions by F. Kahlhoefer,
following Ref. Dobrich:2018jyi . E949 area has been computed by the KLEVER
collaboration and M. Papucci based on E949 data. All other exclusion regions
have been properly re-computed by M. Papucci, following Ref. Dolan:2014ska .

Figure 40: BC10: ALPs with fermion coupling. Current bounds (filled areas) and
medium-far ($\sim$ 10-15 years) prospects for PBC projects (solid lines) for
ALPs with fermion coupling. CHARM and LHCb filled areas have been adapted to
PBC prescriptions by F. Kahlhoefer, following Ref. Dobrich:2018jyi . E949 area
has been computed by the KLEVER collaboration and M. Papucci based on E949
data. All other exclusion regions have been properly re-computed by M.
Papucci, following Ref. Dolan:2014ska .

####  9.4.3 Axion portal with gluon-coupling (BC11)

This benchmark case considers a scenario in which the ALP $a$ only couples to
the gluon field at a scale $\Lambda$ = 1 TeV. One can write down the
corresponding low-energy Lagrangian at the tree level as

${\cal L}={\cal L_{SM}}+{\cal L_{DS}}+a{g^{2}_{s}\over
8f_{G}}G^{b}_{\mu\nu}\tilde{G^{b\;\mu\nu}}.$ (11)

Because the ALP mixes with the neutral pseudoscalar mesons, it is produced in
any process that produces such mesons. Moreover it can be produced also in $B$
mesons decays, as explained in Section 2.1.4. Details about approximations and
assumptions assumed in computing sensitivities for this benchmark case are
reported in Appendices A and B.

Figure 41 shows the current bounds (as coloured filled areas) and the
prospects for PBC projects (solid lines) both on 5- (FASER) and 10-15 years
(CODEX-b, MATHUSLA200, FASER2) timescale. Below the three pion threshold, the
CODEX-b and MATHUSLA200 reach for this benchmark is conditional upon the
eventual detectors being sensitive to the di-photon final state. Production
from K and B decays depend on UV completion and the results shown assume
$\approx[{\rm log\Lambda^{2}_{UV}}/m^{2}_{t}\pm{\mathcal{O}}(1)]\Rightarrow
1$. The CODEX-b curve has been obtained considering B-decays only, hence it is
conservative. NA62++ and SHiP are also expected to be sensitive to this
benchmark case but they did not provide the sensitivity curves on the
timescale of this paper.

Current bounds arise from flavor physics, old beam-dump experiments and LEP
data. A comprehensive reinterpretation of these data has been performed in
Ref. Aloni:2018vki  in the $m_{\pi}<m_{a}<3$ GeV mass region, namely:

  1. 1.

data from LEP Knapen:2016moh ; Abbiendi:2002je  and old beam dump experiments,
E137Bjorken:1988as  and NuCal Blumlein:1990ay , have been used to recast limit
on the $a\gamma\gamma$ vertex and translated into a limit in the
$BR(a\to\gamma\gamma)$;

  2. 2.

the limits on the branching fractions of the decays
$\phi\to\pi\pi\gamma\gamma$ and
$\eta^{\prime}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}\pi^{0}$ Tanabashi:2018oca  are
used to set a limit on the rate of the processes $\phi\to\gamma
a(\pi\pi\gamma)$ and $\eta^{\prime}\to\pi^{+}\pi^{-}a(\pi^{+}\pi^{-}\pi^{0})$,
assuming that all the rate is due to ALPs;

  3. 3.

decays driven by the $b\to sa$ penguin diagram are considered and a recast of
results is performed while analyzing:

     * -

the $m_{\eta\pi\pi}$ spectrum of the decay $B^{\pm}\to
K^{\pm}\eta\pi^{+}\pi^{-}$, interpreted as $B^{\pm}\to
K^{\pm}a(\eta\pi^{+}\pi^{-})$, from Ref. Aubert:2008bk ;

     * -

the $m_{K^{*}K}$ spectrum of the decay $B^{\pm}\to K^{\pm}K^{\pm}K_{\rm
S}\pi^{\mp}$, interpreted as $B^{\pm}=K^{\pm}a(K^{\pm}K_{\rm S}\pi^{\mp})$,
from Ref. Aubert:2008bk ;

     * -

the measurement of the two decay rates, $BR(B^{0}\to K^{0}\phi\phi)$
Lees:2011zh  and $BR(B^{\pm}\to K^{\pm}\omega(3\pi))$ Chobanova:2013ddr , to
put a constraints on the processes $B^{0}\to K^{0}a(\phi\phi)$ and $B^{\pm}\to
K^{\pm}a(\pi^{+}\pi^{-}\pi^{0})$, respectively.

  4. 4.

measurements on processes driven by the $s\to d$ penguin diagram, as
$K^{\pm}\to\pi^{\pm}\gamma\gamma$ Ceccucci:2014oza  and $K_{\rm
L}\to\pi^{0}\gamma\gamma$ Abouzaid:2008xm , are used to recast limits on ALPs.

For cases 2) and 3) listed above, at one loop, the $agg$ vertex generates an
axial-vector $att$ coupling Bauer:2017ris  which enhances the rate for $B\to
K^{(*)}a$ decays Batell:2009jf ; Hiller:2004ii ; Bobeth:2001sq ; Choi:2017gpf
. Following Ref. Aloni:2018vki  the UV-dependent factor contained in the loop,
$\approx[{\rm log\Lambda^{2}_{UV}}/m^{2}_{t}\pm{\mathcal{O}}(1)]$, is
approximated to unity (which corresponds to a UV scale $\sim$ TeV).

Figure 41: Current bounds (as coloured filled areas) and the prospects for PBC
projects (solid lines) both on 5- (FASER) and 10-15 year (CODEX-b,
MATHUSLA200, FASER2) timescale. The CHARM gray filled area has been computed
by F. Kling, recasting the search for long-lived particles decaying to two
photons performed at CHARM Bergsma:1985qz . Other coloured filled areas are
kindly provided by Mike Williams and revisited from Ref. Aloni:2018vki . The
gray areas depend on UV completion and the results shown assume $\approx[{\rm
log\Lambda^{2}_{UV}}/m^{2}_{t}\pm{\mathcal{O}}(1)]\Rightarrow 1$. The CODEX-b
curve has been obtained considering B-decays only, hence it is conservative.
Both NA62++ and SHiP are sensitive to this benchmark case too, the curves are
currently being compiled.

##  10 Physics reach of PBC projects in the multi-TeV mass range

The PBC projects have sensitivity to physics beyond the Standard Model at and
above the TeV mass scale. Since the center of mass energy in the collisions
for the PBC experiments is small compared to the LHC experiments, this
sensitivity comes through modifications of known particle properties through
virtual exchanges of New Physics particles. In some cases, when new physics
violates exact or approximate symmetries of the SM (such as $CP$ symmetry,
and/or lepton flavour), the standard model backgrounds are very low. As a
result precision measurements can be sensitive to NP in the multi-TeV range.

###  10.1 Measurement of EDMs as probe of NP in the multi TeV scale

Measurements of, and constraints on, Electric Dipole Moments (EDMs) of
elementary particles and atoms are a very powerful way of probing theories of
New Physics. One of the key puzzles of the Standard Model, the smallness of
$CP$-violation in the QCD sector, originates from the tight bounds on neutron
and atomic EDMs. The axion solution to this strong $CP$ problem implies the
existence of heavy Peccei-Quinn sectors at high-energy scales that cannot be
directly accessed in high-energy experiments, and so an alternative approach
is needed to understand the solution to this problem. New physics at the weak
scale (or more generically, TeV scale and beyond) physics can also induce
EDMs. Famously, the Kobayashi-Maskawa $CP$ violation mechanism does not induce
neutron or proton EDMs above $10^{-32}e{\rm cm}$, which is firmly outside the
reach of current and next generation EDM experiments. This opens up the
possibility of exploring the TeV frontier with EDMs by increasing the
experimental sensitivity. One of the long-term proposals to measure EDMs is
the proton (and other charged nuclei) storage ring where EDMs can be probed to
unprecedented precision.

If new $CP$-violating physics is heavy, for the purpose of the EDM
description, one can encapsulate its effect in form of the SM effective
operators. For example, the following operators can be interpreted as up- and
down- quark electric dipole moments $d_{u(d)}$:

${\cal
L}=\frac{v\times\sin(\phi^{(u)})}{\Lambda_{u}^{2}}\times\frac{ie}{2}\bar{u}F_{\mu\nu}\sigma_{\mu\nu}\gamma_{5}u+\frac{v\times\sin(\phi^{(d)})}{\Lambda_{d}^{2}}\times\frac{ie}{2}\bar{d}F_{\mu\nu}\sigma_{\mu\nu}\gamma_{5}d+...$
(12)

The insertion of the SM vacuum expectation $v=246$ GeV is necessitated by the
$SU(2)\times U(1)$ gauge invariance. Possible small Yukawa couplings, loop
factors etc have been subsumed into energy scale coefficients $\Lambda_{u}$
and $\Lambda_{d}$. $\phi^{(d)}$ and $\phi^{(u)}$ indicate CP-violating phases.

There is, of course, a wide variety of possible dimension six operators, and
more independent EDM measurements and constraints are required to limit them
all. Non-perturbative methods can be used to relate proton/neutron EDMs to the
quark EDM and other $CP$-odd operator coefficients. Suppressing $u,d$ flavour
dependence, and taking for simplicity $d_{p}\sim{\mathcal{O}}(d_{q})$, one
arrives at the maximum expected sensitivity of protons EDM being reinterpreted
as the sensitivity to $\Lambda$,

$\frac{|\sin(\phi^{(q)})|}{\Lambda_{q}^{2}}\sim\frac{1}{(7\times 10^{5}\,{\rm
TeV})^{2}}\times\left(\frac{d_{p}}{10^{-29}e{\rm cm}}\right)^{1/2}\;.$ (13)

It is likely that operators in (12) are proportional to small Yukawa couplings
and a loop factor, so that the sensitivity to the actual energy scales of new
physics are several orders of magnitude lower than this estimate indicates.
Even then, the suggested target of $10^{-29}e{\rm cm}$ for $d_{p}$ will cover
models with CP-violation in the multi-100 TeV range, thereby exploring, for
example, the scalar quark mass range which would be expected if the measured
value of the Higgs mass, $m_{h}\simeq 125$ GeV, is interpreted within SUSY
models.

###  10.2 Experiments sensitive to Flavour Violation

Of particular interest for the PBC program is the search for flavour violation
which is almost entirely absent in the SM, but introduced in many beyond the
SM scenarios, including theories with supersymmetry. Two PBC experiments aim
to explore the sensitivity of flavour-violating processes to TeV scale
physics, TauFV and KLEVER.

TauFV
In case of the TauFV experiment, the physics goal is to observe and measure,
or alternatively set the upper bound on, the several lepton-flavour-violating
(LFV) $\tau$ or $D$-meson decays. To estimate the sensitivity to a multi-TeV
New Physics scale, the LFV process $\tau^{\pm}\to\mu^{+}\mu^{\pm}\mu^{-}$ is
considered. Such process is almost entirely forbidden in the Standard Model,
and any attempt to measure this small branching will automatically probe the
New Physics that violates approximate $\tau$ and $\mu$ flavour conservation.

To quantify the New Physics reach, one can introduce a series of effective
operators that mediate such transitions. For this particular decay process, at
lowest order, one can have

${\cal
L}=\frac{e^{i\phi}}{\Lambda_{\mu\tau}^{2}}\times(\bar{\mu}\gamma_{\alpha}\mu)(\bar{\mu}\gamma_{\alpha}\tau)\leavevmode\nobreak\
+(h.c.)+\leavevmode\nobreak\ {\rm other\leavevmode\nobreak\
Lorentz\leavevmode\nobreak\ structures}.$ (14)

In this expression, all coupling constants have been subsumed into the
definition of the energy scale $\Lambda_{\mu\tau}$, apart from a possible
phase $\phi$.

The resulting branching ratio for the $\tau\to 3\mu$ decay in the $m_{\mu}\ll
m_{\tau}$ limit is given by

$\Gamma_{\tau\to 3\mu}=\frac{m_{\tau}^{5}}{256\pi^{3}\Lambda_{\mu\tau}^{4}}.$
(15)

Given the stated goal of the TauFV experiment (in the absence of positive
signal) is to reach the exclusion level of ${\rm BR}_{\tau\to 3\mu}\sim
10^{-10}$, one can translate the above formula to $\Lambda_{\mu\tau}$
sensitivity,

$\Lambda_{\mu\tau}>55\,{\rm TeV}\times\left(\frac{10^{-10}}{{\rm Br}_{\tau\to
3\mu}}\right)^{1/4}.$ (16)

KLEVER
The KLEVER proposal seeks to complement the NA62 experiment by measuring
$K_{L}\to\pi^{0}+\,\mbox{missing\,energy}$. In the Standard Model, the missing
energy is carried by neutrinos, $K_{L}\to\pi^{0}\nu\bar{\nu}$, and the
corresponding branching ratio is predicted to be ${\rm BR_{SM}}=(3.4\pm
0.6)\times 10^{-11}$ Buras:2015qea .

The branching ratios for the decays $K\to\pi\nu\bar{\nu}$ are among the
observables in the quark-flavor sector most sensitive to NP. Because the SM
decay amplitudes are strongly suppressed by the GIM mechanism and the CKM
hierarchy and dominated by short-distance physics, the SM rates are small and
predicted very precisely, making the $K\to\pi\nu\bar{\nu}$ BRs potentially
sensitive to NP at mass scales of hundreds of TeV, in general surpassing the
sensitivity of $B$ decays in SM extensions Buras:2014zga . Observations of
lepton-flavor-universality-violating phenomena are mounting in the $B$ sector.
Most explanations for such phenomena predict strong third-generation couplings
and thus significant changes to the $K\to\pi\nu\bar{\nu}$ BRs through
couplings to final states with tau neutrinos Bordone:2017lsy .

The BR for the decay $K_{L}\to\pi^{0}\nu\bar{\nu}$ has never been measured.
The current experimental result, ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=1.73^{+1.15}_{-1.05}\times 10^{-10}$,
obtained at Brookhaven from $K^{+}$ decays at rest with seven candidate
events, Artamonov:2009sz , together with considerations of isospin symmetry
Grossman:1997sk , leads to the model-independent bound ${\rm
BR}(K_{L}\to\pi^{0}\nu\bar{\nu})<1.4\times 10^{-9}$. This limit has to be
compared to the direct limit set by the KOTO experiment, ${\rm
BR}(K_{L}\to\pi^{0}\nu\bar{\nu})<1.4\times 10^{-9}$ at 90 % CL Ahn:2018mvc .
Because the amplitude for $K^{+}\to\pi^{+}\nu\bar{\nu}$ has both real and
imaginary parts while the amplitude for $K_{L}\to\pi^{0}\nu\bar{\nu}$ is
purely imaginary, the decays have different sensitivity to new sources of $CP$
violation.

In general, the measurement of the $BR(K_{L}\to\pi^{0}\nu\overline{\nu}$) is
sensitive to additional sources of flavour violation coming from NP at, or
above, the TeV scale. Parametrizing the effective Lagrangian for new physics
in terms of effective operators as before, and taking one flavour of neutrinos
for simplicity,

${\cal
L}=\frac{e^{i\phi}}{\Lambda_{ds}^{2}}\times(\bar{\nu}\gamma_{\alpha}(1-\gamma_{5})\nu)(\bar{d}\gamma_{\alpha}(1-\gamma_{5})s)\leavevmode\nobreak\
+(h.c.)+\leavevmode\nobreak\ {\rm other\leavevmode\nobreak\
Lorentz\leavevmode\nobreak\ structures},$ (17)

one can quantify the sensitivty of KLEVER to NP. The decay
$K_{L}\to\pi^{0}\nu\bar{\nu}$ is $CP-$violating, and therefore the amplitude
is proportional to $\sin(\phi)$, being $\phi$ the phase of NP contributions.
In contrast, the $K^{+}\to\pi^{+}\nu\bar{\nu}$ branching fraction is phase-
independent, so it can also be used as a probe of TeV physics independently
from $K_{L}\to\pi^{0}\nu\bar{\nu}$. Should NA62 discover deviations from the
Standard Model, a KLEVER-type of measurement would be required to further
investigate their origins.

If the sensitivity of KLEVER can reach the SM branching ratio level, it would
also entail sensitivity to New Physics at approximately

$\frac{|\sin\phi|}{\Lambda_{ds}^{2}}\sim\frac{1}{(500\,{\rm TeV})^{2}}.$ (18)

Figure 42, reproduced from Buras:2015yca , illustrates a general scheme for
the expected correlation between the charged and neutral decays under various
scenarios.

Figure 42: Scheme for BSM modifications of $K\to\pi\nu\bar{\nu}$ BRs.
Reproduced from Ref. Buras:2015yca .

If the NP has a CKM-like structure of flavor interactions, the $K_{L}$ and
$K^{+}$ BRs will lie along the band of correlation shown in green. In models
with only left-handed or only right-handed couplings to the quark currents
(e.g., models with modified $Z$ couplings or littlest-Higgs models with $T$
parity), because of constraints from $\epsilon^{\prime}_{K}$, the BRs must lie
along one of the branches shown in blue. If the NP has an arbitrary flavor
structure and both left-handed and right-handed couplings (e.g., in Randall-
Sundrum models), there is little correlation, as illustrated in red.

In a recent breakthrough, the RBC-UKQCD Collaboration obtained the first
result for ${\rm Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$ from a lattice
calculation thought to have reliable systematics: ${\rm
Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}=(1.38\pm 5.15\pm 4.59)\times 10^{-4}$,
2.1 $\sigma$ less than the experimental value Bai:2015nea . Estimates from
large-Nc dual QCD support the lattice result Buras:2015xba . With this result
for ${\rm Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$, the correlation between
$\epsilon_{K}$ and ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ has been examined
in various SM extensions at energy scales $\Lambda$ in the neighborhood of
1–10 TeV by several authors, in many cases, with constraints from
$\epsilon^{\prime}_{K}$, $\Delta m_{K}$, and ${\rm BR}(K_{L}\to\mu\mu)$
considered as well. The results of these studies are summarized in Table 7. In
general, an observed value of $\epsilon_{K}$ that is larger than expected in
the SM implies a suppression of ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ to
below the SM value. However, it is possible to construct models in which
$\epsilon_{K}$ and ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ are simultaneously
enhanced. With moderate parameter tuning (e.g., cancellation among SM and NP
interference terms to the 10–20% level), ${\rm
BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ may be enhanced by up to an order of
magnitude.

Model |  $\Lambda$ [TeV] |  Effect on ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ |  Effect on ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ | Refs.
---|---|---|---|---
Leptoquarks, most models | 1–20 | Very large enhancements; mainly ruled out | Bobeth:2017ecx
Leptoquarks, $U_{1}$ | 1–20 | +10% to +60% | +100% to +800% | Bobeth:2017ecx
Vector-like quarks | 1–10 |  $-90$% to +60% |  $-100$% to +30% | Bobeth:2016llm
Vector-like quarks + $Z^{\prime}$ | 10 |  $-80$% to +400% |  $-100$% to 0% | Bobeth:2016llm
Simplified modified $Z$, no tuning | 1 |  $-100\%$ to +80% |  $-100$% to $-50$% | Endo:2016tnu
General modified $Z$, cancellation to 20% | 1 |  $-100$% to +400% |  $-100$% to +500% | Endo:2016tnu
SUSY, chargino $Z$ penguin | 4–6 TeV |  |  $-100$% to $-40$% | Endo:2016aws
SUSY, gluino $Z$ penguin | 3–5.5 TeV | 0% to +60% |  $-20$% to +60% | Endo:2017ums
SUSY, gluino $Z$ penguin | 10 | Small effect | 0% to +300% | Tanimoto:2016yfy
SUSY, gluino box, tuning to 10% | 1.5–3 |  $\pm 10$% |  $\pm 20$% | Crivellin:2017gks
LHT | 1 |  $\pm 20$% |  $-10$% to $-100$% | Blanke:2015wba
Table 7: Effects on BRs for $K\to\pi\nu\bar{\nu}$ decays in various SM
extensions, with constraints from other kaon observables, including in
particular ${\rm Re}\,\epsilon^{\prime}_{K}/\epsilon_{K}$.

The KLEVER experiment aims to use a high-energy neutral beam at the CERN SPS
to achieve 60-event sensitivity for the decay $K_{L}\to\pi^{0}\nu\bar{\nu}$ at
the SM BR with an $S/B$ ratio of 1. At the SM BR, this would correspond to a
relative uncertainty of about 20%, demonstrating a discrepancy with $5\sigma$
significance if the observed rate is a bit more than twice or less than one-
quarter of the SM rate, or with $3\sigma$ significance if the observed rate is
less than half of the SM rate. These scenarios are consistent with the rates
predicted for many different SM extensions, as seen in Table 7.

###  10.3 $B$ physics anomalies and $BR(K\to\pi\nu\bar{\nu})$

A number of anomalies have been observed, some of which are $3\sigma$
deviations from Standard Model predictions, in semi-leptonic $B$ decays
Aaij:2014ora ; Aaij:2017vbb . The upcoming analysis of the full LHC Run II
data set (as well as future Belle II experiments) will go a long way towards
clarifying the status of these anomalies: Are they evidence for new physics,
or just statistic fluctuations? Taken together the anomalies hint at a
violation of Lepton Flavour Universality. PBC experiments such as NA62 and
KLEVER can therefore shed complementary light on explanations for these
anomalies. Explanations for the B anomalies include models with flavour
violation only in the third generation Bordone:2017lsy , theories with an
additional $Z^{\prime}$ Kamenik:2017tnu , and theories with leptoquarks
Barbieri:2015yvd .

In most such models the decay $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$, as probed
by KLEVER, could be as sensitive to the physics responsible to the anomalies
as $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$. The key question then is to which
level of precision can one measure these branching ratios relative to the SM
expectation and the timescale within which such sensitivity can be reached.
With the sensitivities discussed for the PBC program, both NA62 and KLEVER can
shed light on many of the possible explanations for the anomalies.

##  11 Conclusions and Outlook

In the past decade, one of the major accomplishments of particle physics has
been the discovery of the Higgs boson that has succesfully completed the
experimental validation of the SM. Beyond this outstanding achievement, a
wealth of experimental results have been produced by the ATLAS, CMS and LHCb
collaborations during Run 1 and Run 2 of the LHC: these collaborations have
explored in depth the paradigm of NP at the TeV scale, required to solve the
hierarchy problem in case of the presence of an intermediate scale between the
EW and the Planck scales. The search for NP has been performed so far both via
direct searches and through precision measurements in flavor: tremendous
progress has been achieved in understanding the SM structure in the last
decades.

This progress is expected to continue for the next two decades: the upgrade of
the LHCb experiment in 2019-2020 will allow a dataset of 50 fb-1 to be
collected in about five years of operation. Major upgrades of the ATLAS and
CMS detectors are also scheduled in 2023-2026 with the ultimate aim to reach
an integrated luminosity of about 3000 fb-1 by around 2035.

Away from the LHC, Belle II is expected to collect an integrated luminosity of
50 ab-1 by 2024. This will provide a dataset that is about a factor of 50
times larger than that collected by BaBar and Belle in the recent past. The
Mu2E experiment at FNAL, and the Mu3e and the upgrade of MEG experiments at
the PSI in the next decade will advance tremendously in the investigation of
NP in charged lepton-flavor-violating processes, nicely completing and
complementing the quest of NP perfomed at the LHC experiments and at the
B-factory.

However, the absence, so far, of unambiguous signal of NP from direct searches
at the LHC, indirect searches in flavour physics and direct detection Dark
Matter experiments, along with the absence of a clear guidance from the theory
about the NP scale, imposes today, more than ever, to broadening the
experimental effort in the quest for NP and exploring different ranges of
interaction strengths and masses with respect to what is already covered by
existing or planned in itiatives.

The CERN laboratory could offer an unprecendented variety of high-intensity,
high-energy beams and scientific infrastructures that could be exploited to
this endevour. This effort would nicely complement and further broaden the
already rich physics programme ongoing at the LHC and HL-LHC.

The proposals presented in the PBC-BSM context can search for NP in a fully
complementary range of masses and couplings with respect to those investigated
at the LHC: new particles with masses in the sub-eV range and very weakly
coupled to the SM particles, can be explored by the IAXO and JURA proposals or
through the investigation of oscillating EDMs in protons or deuterons in a
electrostatic ring (CP-EDM); MeV-GeV hidden-sector physics can be explored by
a multitude of experiments at the PS beam lines (REDTOP proposal), SPS beam
lines (NA62++, NA64++, SHiP, KLEVER, LDMX, and NA64/AWAKE proposals) and at
the LHC interaction points (FASER, CODEX-b, MATHUSLA200, and milliQan
proposals). The multi-TeV mass range ($\sim 100$ TeV) can be indirectly
explored both via ultra-rare or forbidden decays (KLEVER and TauFV) and
through the search for a permanent EDM in protons/deuterons (CP-EDM) or in
strange/charmed baryons (LHC-FT).

The Collaborations behind these proposals are backed up by a lively
phenomenological and theoretical community, and represent a fertile ground
where New Physics models can be developped, discussed, and further improved.
These proposals will possibly compete with similar proposals planned in the
world (as, for example, at Jefferson Lab, FNAL, JPARC, KEK, Mainz, PSI, etc)
and complement the current effort in the search for NP in other domains (as,
for example, DM direct searches at Gran Sasso Laboratory, SNOLAB or
elsewehere). They will further enrich the ongoing effort at the LHC to
discover NP at the TeV scale, increasing the impact that CERN could have in
the next 10-20 years on the international landscape.

## Acknowledgements

We are grateful to the colleagues of the Beam Dump Facility, Conventional
Beams, AWAKE and eSPS Physics Beyond Colliders working groups for useful
discussions and synergies. We warmly thank F. Kahlhoefer and M. Williams for
help with recasting some old experimental results related to ALPs with fermion
and gluon couplings. We are grateful to Dean Robinson and Harikrishnan Ramani
for help with computing the CODEX-b reach, and to Marat Freytsis for helpful
discussions related to the penguin calculations for the benchmark cases BC10
and BC11. Individual authors of this report have received financial support
from ERC Ideas Consolidator Grant No. 771642 SELDOM (European Union); MINECO
and GVA (Spain).

##  Appendix A ALPS: prescription for treating the FCNC processes

The prescription for treating the FCNC processes for ALPS production and
consequent decay described below assume a certain number of approximations.
For an in-depth study of ALP production and decay, see the recent work
Aloni:2018vki  where the non-perturbative aspects of the problem are treated
using the data-driven approach, derived from meson production, interaction and
decay.

ALPs with fermion coupling (BC10)
There is a certain degree of UV dependence associated with the production
through $B$-meson decays, and the PBC recommends following the prescription in
Batell:2009jf . Concretely, the effective $b-s-a$ vertex upon integrating out
the $W$ and $t$ is taken to be

$\mathcal{L}\supset\frac{a}{f_{q}}\,\bar{s}_{L}b_{R}\times\frac{\sqrt{2}G_{F}m_{t}^{2}m_{b}V^{\ast}_{ts}V_{tb}}{8\pi^{2}}\times
c^{(BC10)}_{fcnc}+(h.c.).$ (19)

and coefficient $c^{(BC10)}_{fcnc}$ is chosen to be

$c^{(BC10)}_{fcnc}=\log\left(\frac{\Lambda_{UV}^{2}}{m_{t}^{2}}\right).$ (20)

where the threshold (model dependent finite pieces) cannot be determined
without UV completion and are dropped. The generalization to
$\bar{d}_{L}s_{R}a$ interactions is done by taking $V^{\ast}_{ts}V_{tb}\to
V^{\ast}_{td}V_{ts}$.

Taking $\Lambda_{UV}=1$ TeV and again following Batell:2009jf , the branching
ratios are

$\displaystyle\mathrm{Br}(B\to Ka)$ $\displaystyle\approx 1.5\times
10^{-5}\times\left(\frac{100\,\mathrm{TeV}}{f_{q}}\right)^{2}\times(\mathcal{F}_{K}(m_{a}))^{2}\,\lambda_{Ka}^{1/2}$
(21) $\displaystyle\mathrm{Br}(B\to K^{\ast}a)$ $\displaystyle\approx
1.8\times
10^{-5}\times\left(\frac{100\,\mathrm{TeV}}{f_{q}}\right)^{2}\times(\mathcal{F}_{K^{\ast}}(m_{a}))^{2}\,\lambda_{K^{\ast}a}^{3/2}$
(22)

with form-factors extracted from $B$-physics literature:

$\displaystyle\mathcal{F}_{K}(m_{a})$
$\displaystyle=\frac{1}{1-m_{a}^{2}/(38\,\mathrm{GeV}^{2})}$ (23)
$\displaystyle\mathcal{F}_{K^{\ast}}(m_{a})$
$\displaystyle=\frac{3.65}{1-m_{a}^{2}/(28\,\mathrm{GeV}^{2})}-\frac{2.65}{1-m_{a}^{2}/(37\,\mathrm{GeV}^{2})}$
(24) $\displaystyle\lambda_{ij}$
$\displaystyle=\left(1-\frac{(m_{i}+m_{j})^{2}}{m_{B}^{2}}\right)\left(1-\frac{(m_{i}-m_{j})^{2}}{m_{B}^{2}}\right).$
(25)

For the inclusive rate, we assume

$\mathrm{Br}(B\to X_{s}a)\approx 5\times\Big{(}\mathrm{Br}(B\to
Ka)+\mathrm{Br}(B\to K^{\ast}a)\Big{)},$ (26)

following arguments presented in Ref. Aloni:2018vki .

ALPs with gluon coupling (BC11)
In this benchmark case, the ALP can be produced directly in the hadronization
process or through a $B$-meson decay. Concretely, 1-loop operator mixing
generates the effective coupling of already considered in BC10:

$\mathcal{L}\supset\delta
c_{qq}\frac{\partial_{\mu}a}{f_{G}}\sum_{\beta}\bar{q}_{\beta}\gamma_{\mu}\gamma_{5}q_{\beta}$
(27)

where $\delta c_{qq}$ is generated through a gluon loop. The corresponding
log-enhanced coefficient can be found in Ref. Bauer:2017ris .

A full calculation would require specifying a UV model, especially since the
log-enhanced coefficient is not parametrically large. For concreteness we
follow the choice in Aloni:2018vki  by (1) dropping all logs and (2) setting
$\delta c_{qq}$ equal to the coefficient of the leading log of the diagram
which generates (27). In formulas this corresponds to

$\mathcal{L}\supset\frac{a}{f_{q}}\,\bar{s}_{L}b_{R}\times\frac{\sqrt{2}G_{F}m_{t}^{2}m_{b}V^{\ast}_{ts}V_{tb}}{8\pi^{2}}\times
c^{(BC11)}_{fcnc}+(h.c.).$ (28)

Out choice for $c^{(BC11)}_{fcnc}$ is

$c^{(BC11)}_{fcnc}\simeq\alpha_{s}^{2}(m_{t}).$ (29)

We note here that there is a significant UV-completion dependence, and further
work would be required to properly estimate the preferred range for
$c^{(BC11)}$.

With this convention, again following Batell:2009jf , the branching fractions
are

$\displaystyle\mathrm{Br}(B\to Ka)$ $\displaystyle\approx 6.6\times
10^{-10}\times\left(\frac{100\,\mathrm{TeV}}{f_{G}}\right)^{2}\times(\mathcal{F}_{K}(m_{a}))^{2}\,\lambda_{Ka}^{1/2}$
(30) $\displaystyle\mathrm{Br}(B\to K^{\ast}a)$ $\displaystyle\approx
7.9\times
10^{-10}\times\left(\frac{100\,\mathrm{TeV}}{f_{G}}\right)^{2}\times(\mathcal{F}_{K^{\ast}}(m_{a}))^{2}\,\lambda_{K^{\ast}a}^{3/2}.$
(31)

Again, the uncertainty in the amplitude could result in as much as
$\mathcal{O}(10)$ changes in the width, with more accurate calculation of RG
effects and threshold corrections in UV complete models.

Approximation for ALP lifetime

For computing the ALP lifetime the PBC has taken the following approximations,
depending on the mass range considered.

  * •

Region 1, $m_{a}<3m_{\pi}$, photon decay $a\to\gamma\gamma$. In this case the
decay is dominated by the two-photon contribution.

$\displaystyle\Gamma_{\rm
tot}(m_{a}<3m_{\pi})\approx\Gamma_{\gamma\gamma}=\frac{m_{a}^{3}}{f_{G}^{2}}\times\frac{\pi\alpha^{2}}{4}\times\left(\frac{4m_{d}+m_{u}}{3(m_{u}+m_{d})}-\frac{m_{a}^{2}}{m_{a}^{2}-m_{\pi}^{2}}\frac{m_{d}-m_{u}}{2(m_{u}+m_{d})}\right)^{2}$
(32)
$\displaystyle=\frac{m_{a}^{3}}{f_{G}^{2}}\times\frac{\pi\alpha^{2}}{4}\times\left(1.0-0.18\times\frac{m_{a}^{2}}{m_{a}^{2}-m_{\pi}^{2}}\right)^{2}.$
(33)

Notice that the $\frac{4m_{d}+m_{u}}{3(m_{u}+m_{d})}$ ratio in (32), to good
accuracy, is 1 for the physical point of $m_{u}/m_{d}\simeq 0.48$. The second
term in the bracket comes from the mixing with $\pi^{0}$ meson, see Ref.
Bauer:2017ris . This formula can be further improved by including
contributions from mixing with $\eta$ (and $\eta^{\prime}$).

  * •

Region 2, $3m_{\pi}<m_{a}<2m_{\pi}+m_{\eta}$, $a\to 3\pi$ decay.

In this region, two new decay modes, $\pi^{+}\pi^{-}\pi^{0}$ and $3\pi^{0}$,
open up. Within 2-flavour chiral perturbation theory, these decays were
treated in Bauer:2017ris . The results are chirally suppressed, $\Gamma_{a\to
3\pi}\propto m_{a}m_{\pi}^{2}/(F_{\pi}f_{G})^{2}$. Using formulae from
Bauer:2017ris , we adopt it to normalization used in these notes,

$\displaystyle\Gamma_{tot}=\Gamma_{\gamma\gamma}+\Gamma_{a\to
3\pi};\leavevmode\nobreak\ \leavevmode\nobreak\ \Gamma_{a\to
3\pi}=\frac{\pi}{3\times
128}\frac{m_{a}m_{\pi}^{4}}{F_{\pi}^{2}f_{G}^{2}}\left(\frac{m_{a}^{2}}{m_{a}^{2}-m_{\pi}^{2}}\frac{m_{d}-m_{u}}{m_{u}+m_{d}}\right)^{2}\times
I(m_{\pi}^{2}/m_{a}^{2}),$ (34) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
I(y)=\int_{4y}^{(1-\sqrt{y})^{2}}\sqrt{1-\frac{4y}{x}}\sqrt{(1-x-y)^{2}-4xy}\times[12(x-y)^{2}+2].$

One can check that $\Gamma_{a\to 3\pi}$ is comparable to
$\Gamma_{\gamma\gamma}$. Asymptotically 212121This formula is valid in the
regime of $f_{G}\gg F_{\pi}$, $\theta_{\pi}\ll 1$., at large $m_{a}$, $I\to
2$. Given the experience with $\eta$ decays Gasser:1984pr , the validity of
the leading chiral order answer is within a factor of $\sim 3$, and can be
improved by including next orders in the chiral expansion. In this mass
region, $a\to\eta^{*}\to 3\pi$ mediated decay is also important, especially
near $m_{a}=m_{\eta}$.

  * •

Region 3, $2m_{\pi}+m_{\eta}<m_{a}<1.5\,{\rm GeV}$, multiple hadronic decays.
Above the $2m_{\pi}+m_{\eta}\sim 830$ MeV threshold many new hadronic
contributions open up, ($\eta\pi\pi$, $\rho\pi$, also $f_{0}\pi$ etc) so that
the result is much larger than chiral perturbaton theory answer for $a\to
3\pi$. One could use hadronic resonance models to have a phenomenological
description of $a$ decays, but for sake of simplicity we suggest an
interpolating formula for the $a$ decay. The following formula,

$\Gamma_{\rm tot}(m_{1}<m_{a}<m_{2})\approx
A(m_{a}-B)^{3};\leavevmode\nobreak\
A=\frac{\Gamma_{2}(1-r)^{3}}{(m_{2}-m_{1})^{3}};\leavevmode\nobreak\
B=\frac{m_{1}-rm_{2}}{1-r};\leavevmode\nobreak\
r=(\Gamma_{1}/\Gamma_{2})^{1/3}$ (35)

interpolates in the region from $m_{1}=2m_{\pi}+m_{\eta}$ to $m_{2}=1.5$ GeV
where $\Gamma_{1}=\Gamma_{\gamma\gamma}(m_{1})+\Gamma_{3\pi}(m_{1})$ and
$\Gamma_{2}=\Gamma_{gg}(m_{a}=1.5\,{\rm GeV})$ is the inclusive decay to
gluons (see below). This interpolation captures the rapidly growing decay rate
by introducing $\propto m_{a}^{3}$ scaling. $m_{a}=1.5$ GeV is chosen to be
the lower boundary of the perturbative description, and this choice bears
significant uncertainty.

  * •

Region 4, $m_{a}>1.5\,{\rm GeV}$, perturbative description. At $m_{a}\sim
1.1-1.5$ GeV, many new additional hadronic decays of $a$ open up,
$\pi_{0}f_{0}(980)$, $\pi a_{0}$, $\eta f_{0}$, $\rho\rho$, $KK^{*}$ etc,
quickly driving up the value for $\Gamma_{\rm tot}$. Asymptotically, the sum
of all hadronic states approaches the perturbative $a\to gluons$ answer. PBC
recommends using the the perturbative formula of $a$ decays to gluons as a
proxy for hadronic decays above $1.5$ GeV:

$\Gamma_{\rm tot}(m_{a}>1.5\,{\rm
GeV})\approx\Gamma_{gg}=\frac{m_{a}^{3}}{f_{G}^{2}}\times\frac{\pi\alpha^{2}_{s}(m_{a})}{2},$
(36)

This is an order-of-magnitude estimate, that cannot be improved using
pertubation theory, and may only be improved with non-perturbative methods.

  * •

Resonance regions, $m_{a}\sim m_{\eta}$, $m_{a}\sim m_{\eta^{\prime}}$.

In addition to the above expressions, one needs to add strong resonant
contributions when $m_{a}$ becomes close to $m_{\eta}$ and
$m_{\eta^{\prime}}$. If the continuum - resonance interference is neglected,
this is achieved via the following formulae,

$\displaystyle\Gamma_{a-\eta,res}=(2\pi^{2}\cos\theta_{p})^{2}\times\frac{F_{\pi}^{2}}{f_{G}^{2}}\times\frac{m_{a}^{4}\Gamma_{\eta}(m_{a})}{(m_{a}^{2}-m_{\eta}^{2})^{2}+m_{\eta}^{2}\Gamma_{\eta}^{2}(m_{a})},\leavevmode\nobreak\
\leavevmode\nobreak\ $ (37)
$\displaystyle\Gamma_{a-\eta^{\prime},res}=(2\pi^{2}\sin\theta_{p})^{2}\times\frac{F_{\pi}^{2}}{f_{G}^{2}}\times\frac{m_{a}^{4}\Gamma_{\eta^{\prime}}(m_{a})}{(m_{a}^{2}-m_{\eta^{\prime}}^{2})^{2}+m_{\eta^{\prime}}^{2}\Gamma_{\eta^{\prime}}^{2}(m_{a})},$
(38)

where $\Gamma_{\eta}(m)$ and $\Gamma_{\eta^{\prime}}(m)$ are the energy-
dependent widths $\Gamma(E)$ evaluated at $E=m_{a}$.

Value for mixing angles, $\cos\theta_{p}\simeq 0.6$ and $\sin\theta_{p}\simeq
0.8$, are taken from Ref. Fariborz:1999gr . More details on mixing
coefficients are given in the Appendix.

Theoretical input is required in deriving $\Gamma_{\eta(\eta^{\prime})}(E)$,
where the main effect is due to the available phase space for $3\pi$ and $\eta
2\pi$ final states. PBC suggest using a simple approximate formula that
reflects the growth of phases space. For $\eta$ meson we take

$\displaystyle\Gamma_{\eta}(E)=\Gamma_{\eta}\times\frac{f(E)}{f(m_{\eta})};\leavevmode\nobreak\
\leavevmode\nobreak\
f(E)=\frac{(E-3m_{\pi})^{1.5}}{(E-3m_{\pi})^{1.5}+(300\,{\rm
MeV})^{1.5}}\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\ E>3m_{\pi};$
(39) $\displaystyle f(E)=0\leavevmode\nobreak\ {\rm for}\leavevmode\nobreak\
E<3m_{\pi},$

where $\Gamma_{\eta}$ is the total width of [physical] $\eta$ meson. For
$\eta^{\prime}$, the same formula applies, with
$\Gamma_{\eta}\to\Gamma_{\eta^{\prime}}$ and $3m_{\pi}\to 2m_{\pi}+m_{\eta}$
substitutions. Better descrition can be achieved with the use of hadronic
models.

##  Appendix B ALPs: production via $\pi^{0},\eta,\eta^{\prime}$ mixing

If $m_{a}$ is below the hadronic scale of $\sim 4\pi F_{\pi}$ ($F_{\pi}=93$
MeV), one can neglect heavy flavours and try to use chiral perturbation theory
by replacing $G\tilde{G}$ operator with using the anomaly equation. We use
this equation for three light quarks ($q_{i}=u,d,s$) in the following form,

$\frac{\alpha_{s}}{8\pi}G^{b}\tilde{G}^{b}=\sum_{i}\frac{m_{*}}{2m_{i}}\partial_{\mu}\bar{q}_{i}\gamma_{\mu}\gamma_{5}q_{i}-m_{*}\sum_{i}\bar{q}_{i}i\gamma_{5}q_{i}-\frac{\alpha}{4\pi}F\tilde{F}\sum_{i}\frac{N_{c}Q_{i}^{2}m_{*}}{m_{i}}$
(40)

where we suppress the Lorentz indices over the gluon and photon fields
strength, $G$ and $F$. Here $Q_{i}$ are the quark charges in units of $e$,
$N_{c}=3$ and $m_{*}\equiv(\sum_{i}m_{i}^{-1})^{-1}$. Dropping terms
suppressed by $m_{u(d)}/m_{s}$, we have $m_{*}=m_{u}m_{d}/(m_{u}+m_{d})$ and

$\displaystyle\frac{\alpha_{s}}{8\pi}G^{b}\tilde{G}^{b}=\frac{m_{d}}{2(m_{u}+m_{d})}\partial_{\mu}\bar{u}\gamma_{\mu}\gamma_{5}u+\frac{m_{u}}{2(m_{u}+m_{d})}\partial_{\mu}\bar{d}\gamma_{\mu}\gamma_{5}d-m_{*}\sum_{i}\bar{q}_{i}i\gamma_{5}q_{i}-\frac{\alpha}{4\pi}F\tilde{F}\frac{4m_{d}+m_{u}}{3(m_{u}+m_{d})}$
(41)

In the leading chiral order, the flavour-singlet
$m_{*}\sum_{i}\bar{q}_{i}i\gamma_{5}q_{i}$ combination can be neglected, and
the total Lagrangian at low energy can be rewritten as

${\cal L}_{\rm axion,l.e.}={\cal L}_{\rm SM}+{\cal L}_{\rm
DS}+4\pi^{2}\times\frac{a}{f_{G}}\frac{\alpha}{4\pi}\frac{4m_{d}+m_{u}}{3(m_{u}+m_{d})}F_{\mu\nu}\tilde{F}_{\mu\nu}-4\pi^{2}\times\frac{\partial_{\mu}a}{2f_{G}}\left(J_{A,\mu}^{S}+\frac{m_{d}-m_{u}}{m_{u}+m_{d}}J_{A,\mu}^{T}\right),$
(42)

where $J_{A,\mu}^{S}$ is the singlet,
$\frac{1}{2}(\bar{u}\gamma_{\mu}\gamma_{5}u+\bar{d}\gamma_{\mu}\gamma_{5}d)$,
and $J_{A,\mu}^{T}$ is the triplet axial-vector current,
$\frac{1}{2}(\bar{u}\gamma_{\mu}\gamma_{5}u-\bar{d}\gamma_{\mu}\gamma_{5}d)$.
Interaction with $J_{A,\mu}^{S}$ leads to $a-\eta$ and $a-\eta^{\prime}$
mixing, while interaction with $J_{A,\mu}^{T}$ results in $a-\pi^{0}$ mixing.

Using the model that relates octet and singlet quark states to physical $\eta$
and $\eta^{\prime}$ Fariborz:1999gr , and usual rules for $\langle
0|J_{A,\mu}|pseudoscalar\rangle$ matrix elements, we transform the last term
in (42) to an on-shell mixing between $a$ and the pseudoscalars,

$\displaystyle{\cal L}_{\rm
axion,l.e.}=...4\pi^{2}\times\frac{\partial_{\mu}a}{2f_{G}}\left(J_{A,\mu}^{S}+\frac{m_{d}-m_{u}}{m_{u}+m_{d}}J_{A,\mu}^{T}\right)$
$\displaystyle\to
4\pi^{2}\times\frac{F_{\pi}}{2f_{G}}\left(\partial_{\mu}a\partial_{\mu}\eta\times\cos\theta_{p}+\partial_{\mu}a\partial_{\mu}\eta^{\prime}\times\sin\theta_{p}+\partial_{\mu}a\partial_{\mu}\pi_{0}\frac{m_{d}-m_{u}}{m_{u}+m_{d}}\right),$
(43)

where $\cos\theta_{p}\simeq 0.6$ and $\sin\theta_{p}\simeq 0.8$ related
physical $\eta$ and $\eta^{\prime}$ with octet and singlet combinations
Fariborz:1999gr .

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