# Weighing the Axion with Meson Haloscopy

N. Bray-Ali<EMAIL_ADDRESS>Department of Physical Sciences
Mount Saint Mary’s University
Los Angeles, CA 90049

###### Abstract

Recent results for the optical conductivity of the vacuum at nuclear
frequencies have confirmed and increased the long-standing tension between
probes with pions and measurements with muons. State-of-the-art simulations
with quarks and gluons fall between the muon and pion results. We resolve the
tension using haloscopy: axions in the local dark matter halo of the galaxy
shift the magnetic moment of the muon and collide with charged pions to make
neutral ones. The effects scale with the four-dimensional volume of the
haloscope and show that axions saturate the local halo energy density. Taking
the cosmic dark matter energy density to be similarly saturated by axions, we
find around six axions for each photon and offer a plausible scenario leading
to this ratio.

The standard model of particle physics uses the renormalization group (RG) to
infer the effect of virtual particles and field fluctuations on real particles
and classical fields[1]. By contrast, the standard model of cosmology uses
general relativity (GR) to get the effect of dark matter and dark energy on
normal matter and radiation[2]. In this Letter, we take the cosmological
approach to particle physics and look for the effect of axions in the local
dark matter halo of the galaxy on measurements with muons, pions, and other
particles with mass in between the electron and proton111Haloscopes looking
for the effect of axions on radio-frequency electromagnetic cavities have
yielded important null results in the $\mu{\rm eV}$ range for the axion
mass[18]. The axion mass $m_{A}=(0.492\pm 0.013)~{}{\rm eV/c^{2}}$ that we
obtain from meson haloscopy explains these null results, and suggests we look
instead in the infrared regime at resonant frequency
$\omega_{A}/(2\pi)=(120\pm 3)~{}{\rm THz}$ and vacuum wavelength
$\lambda_{A}=(2.52\pm 0.06)~{}\mu{\rm m}$..

The motivation for this approach comes from the increasing tension between
probes of the optical conductivity of the vacuum at nuclear frequencies using
pions[4], muons[5], and quarks[6]. On the other hand, estimates of the dark
matter density and other cosmological parameters converged almost a decade ago
at the per cent level of precision[7]. We find that we can resolve the tension
and weigh the axion by interpreting these probes as meson haloscopes (See Fig.
1 and Fig. 2). Strikingly, this gives an energy scale $f_{A}=(1.16\pm
0.03)\times 10^{7}~{}{\rm GeV}$ controlling axion decay that lies far below
the scale where the strengths of the nuclear forces are thought[1] to get
close ($10^{16}~{}{\rm GeV}$), yet still far above where the symmetry between
electromagnetism and the weak nuclear force breaks ($10^{2}~{}{\rm GeV}$). We
come back to the physics at this energy after sketching how $f_{A}$ emerges
from meson haloscopy222We find a value for the axion-nucleon coupling
$g=(5.2\pm 0.4)\times 10^{-7}$ using meson haloscopy that excludes the
standard “core collapse” scenario for supernova[28, 29]. Axions with this
value of $g$ thermalize in the putative collapsed core of a supernova
progenitor star and radiate with greater luminosity than the electron anti-
neutrino $\overline{\nu}_{e}$ luminosity ($\approx 10^{53}~{}{\rm erg/sec}$)
inferred from the observed $\overline{\nu}_{e}$ pulse of the 1987 supernova in
the Large Magellanic Cloud[28, 29]. The core would thus cool faster than the
time ($\approx 3~{}{\rm sec}$) it takes for the $\overline{\nu}_{e}$ pulse to
form. We emphasize that there is no standard model for supernova generation
and that the physics at super-nuclear energy density ($\approx 1.7\times
10^{-3}~{}{\rm GeV^{4}}$) thought to be relevant to this important
astrophysical process are largely unknown. For comparison, the topological
susceptibility of the quark-gluon vacuum $\chi=m_{A}^{2}f_{A}^{2}=(3.25\pm
0.06)\times 10^{-5}~{}{\rm GeV^{4}}$ is roughly 50 times smaller[22],
suggesting that quark and gluon field configurations with non-trivial
topology[30] play a crucial role..

Figure 1: Principle of Muon Haloscopy Halo axion $A$ decays to photon pair and
shifts the coupling of positive muon $\mu^{+}$ with magnetic field $B$.

The starting point is the $\rho-\omega$ mixing angle[9]

$\frac{\delta_{\epsilon}}{2}=\frac{\Delta
M_{\omega}}{\Gamma_{\omega}}=\sqrt{\frac{\rho_{A}V_{A\rho}}{\Gamma_{\rho}}}$
(1)

where, $\Delta M_{\omega}$ is the haloscopic shift in the $\omega$ mass,
$\Gamma_{\omega}$ and $\Gamma_{\rho}$ are the decay rates of the $\omega$ and
$\rho$, $V_{A\rho}$ is the haloscope volume, and $\rho_{A}$ is the axion
energy density. The vector mesons $\rho$ and $\omega$ dominate, respectively,
the two-pion $a_{\mu}^{\pi\pi}$[10, *teubner2019, *michel2019] and three-pion
$a_{\mu}^{\pi\pi\pi}$[13] parts of the magnetic moment $a_{\mu}$ of the
muon[14]. The interference between them breaks isospin, $G$-parity, and
charge-parity ($CP$) symmetries[9]. Remarkably, the axion energy density
$\rho_{A}=(0.35\pm 0.10)~{}{\rm GeV/cm^{3}}$ we get using Eq. (1) agrees with
the local halo energy density $\rho_{H}=(0.45\pm 0.17)~{}{\rm
GeV/cm^{3}}$[15]. Thus, axions seem to form the bulk of the dark matter in the
local halo333We take the $\rho-\omega$ mixing angle $\delta_{\epsilon}=(6.1\pm
0.9)~{}{\rm deg}$ from the KLOE detector of the DAFNE collider at Frascati[9].
Similarly, we get the volume
$V_{A\rho}=\sigma_{x}\sigma_{y}\sigma_{z}=1.2\times 10^{-3}~{}{\rm cm^{3}}$
from the DAFNE beam width at the crossing point inside KLOE [31]:
$\sigma_{x}=2~{}{\rm mm},\sigma_{y}=20~{}\mu{\rm m},\sigma_{z}=3~{}{\rm cm}$.
The $\rho$ line-width is $\Gamma_{\rho}=0.1474(5)~{}{\rm GeV}$[32, 18]. .

Figure 2: Principle of Pion Haloscopy Halo axion $A$ collides with charged
pion $\pi^{\pm}$ and converts to neutral pion $\pi^{0}$.

Axions in the local dark matter halo of the galaxy shift the muon magnetic
moment $a_{\mu}$ and its two-pion part $a_{\mu}^{\pi\pi}$ in opposite
directions (Figs. 1 and 2):

$\displaystyle\frac{\Delta a_{\mu}}{a_{\mu}}$ $\displaystyle=$ $\displaystyle
g_{A\gamma\gamma}m_{A}\sqrt{\frac{\rho_{A}V_{A\mu}N_{\mu}}{\Gamma_{\mu}}}$
$\displaystyle\frac{\Delta a^{\rm\pi\pi}_{\mu}}{a_{\mu}^{\pi\pi}}$
$\displaystyle=$
$\displaystyle-\xi_{A\pi}\frac{m_{\pi}^{2}}{f_{\pi}^{2}}\sqrt{\frac{\rho_{A}V_{A\pi}N_{\pi}}{\Gamma_{\pi}}},$
(2)

where, the axion has mass $m_{A}$, photon coupling $g_{A\gamma\gamma}$, and
pion mixing ratio $\xi_{A\pi}$[17]. The pion mass is $m_{\pi}=0.135~{}{\rm
GeV/c^{2}}$, the pion decay constant is $f_{\pi}=0.0923~{}{\rm GeV}$, and the
fine-structure constant is $\alpha=e^{2}/(\hbar c)\approx 1/137$[18]. The
haloscope parameters are the volumes $V_{A\mu}$ and $V_{A\pi}$, the number of
muons $N_{\mu}$ and pions $N_{\pi}$ at the start of haloscopy, the muon decay
rate $\Gamma_{\mu}$ in the haloscope rest frame, and the bandwidth
$\Gamma_{\pi}$ of the pion haloscope444We take the muon haloscope parameters
from Fermilab’s E989 experiment[5] and the pion parameters from the KLOE
detector at the DAFNE collider[31]. The volume of the muon haloscope is [5,
26, 33] $V_{A\mu}=2\pi^{2}R^{3}\sqrt{C_{e}C_{p}}/(n\sqrt{1-n})=(6.29\pm
0.42)\times 10^{3}~{}{\rm cm^{3}}$, where, $R=711~{}{\rm cm}$ is the radius of
the central orbit of muons in the ring, $C_{e}$ is the electric field
correction from the radial width of the muon beam, $C_{p}$ is the pitch
correction from the width of the beam along the magnetic field, $n\approx
4\times(2\omega_{c}-\omega_{\rm BO})/(2\omega_{c})\approx 0.11$ is the index
for the electrostatic quadrupole field, $\omega_{c}/(2\pi)=c/(2\pi
R)=6.72~{}{\rm MHz}$ is the cyclotron frequency, and $\omega_{\rm
BO}/(2\pi)=13.1~{}{\rm MHz}$ is the betatron oscillation frequency. The number
of muons at the start of haloscopy is [5, 26]:
$N_{\mu}=N_{0}e^{-\Gamma_{\mu}t_{\rm fill}}\approx 3100$, where, $N_{0}\approx
5000$ is the number of muons in the ring at the start of the fill, and $t_{\rm
fill}\approx 30~{}\mu{\rm sec}$ is the time it takes to fill the ring. The
muon decay rate in the haloscope rest frame is
$\Gamma_{\mu}=\hbar/(\gamma_{\mu}\tau_{\mu})=1.20\times 10^{-20}~{}{\rm GeV}$,
where, $\gamma_{\mu}\approx p_{\mu}/(m_{\mu}c)=29.3$ is the muon time dilation
factor, $p_{\mu}=3.10~{}{\rm GeV/c}$ is the muon momentum,
$m_{\mu}=0.106~{}{\rm GeV/c^{2}}$ is the muon rest mass, and
$\tau_{\mu}=2.20~{}\mu{\rm sec}$ is the muon life-time in its rest frame[5].
The volume of the pion haloscope is:$V_{A\pi}=V_{A\rho}=1.2\times
10^{-3}~{}{\rm cm^{3}}$. The number of pions at the start of haloscopy is [34,
35]: $N_{\pi}=2(a_{\mu}^{\pi\pi}/\sigma^{\rm
stat}_{a_{\mu}^{\pi\pi}})^{2}=1.23\times 10^{5}$, where, $\sigma^{\rm
stat}_{a_{\mu}^{\pi\pi}}=1.1\times 10^{-10}$ is the statistical uncertainty of
$a_{\mu}^{\pi\pi}=385.1\times 10^{-10}$. The haloscope bandwidth is set by the
light-crossing time of the luminous region[31]:$\Gamma_{\pi}=\hbar
c/\sigma_{\rm min}=9.87\times 10^{-12}~{}{\rm GeV}$, where, $\sigma_{\rm
min}={\rm min}\left\\{\sigma_{x},\sigma_{y},\sigma_{z}\right\\}=\sigma_{y}$
for the KLOE detector at DAFNE..We can take the quark and gluon simulation
value for $a_{\mu}^{\pi\pi}$ as the effective zero point for measuring the
shifts $\Delta a_{\mu}^{\pi\pi}$ and $\Delta a_{\mu}^{\pi\pi}$555The
haloscopic shift of the muon magnetic moment is $\Delta a_{\mu}=a_{\mu}({\rm
Exp})-a_{\mu}({\rm SM})+a_{\mu}^{\pi\pi}({\rm SM})-a_{\mu}^{\pi\pi}({\rm
Lattice})=(10.6\pm 6.7)\times 10^{-10}$, where, $a_{\mu}({\rm
Exp})-a_{\mu}({\rm SM})=25.1\pm 5.9)\times 10^{-10}$ is the shift of the world
average muon measurement $a_{\mu}({\rm Exp})$[5] from the Standard Model value
$a_{\mu}({\rm SM})$[4]. Here, $a_{\mu}^{\pi\pi}({\rm SM})=(693.1\pm 4.0)\times
10^{-10}$ is the measured two-pion part of the muon magnetic moment[4, 12, 11,
10] and $a_{\mu}^{\pi\pi}({\rm Lattice})=(707.5\pm 5.5)\times 10^{-10}$ is the
simulation result found using quark chromodynamics[6]. Similarly, the
haloscopic shift of the two-pion part is $\Delta
a_{\mu}^{\pi\pi}=a_{\mu}^{\pi\pi}({\rm Exp})-a_{\mu}^{\pi\pi}({\rm
Lattice})=-(18.7\pm 5.8)\times 10^{-10}$, where, we use the KLOE detector at
the DAFNE collider experimental result[10, 4] for $a_{\mu}^{\pi\pi}({\rm
Exp})=(688.8\pm 1.8)\times 10^{-10}$. , since they probe a four-volume
($10^{-39}~{}{\rm cm^{3}/GeV^{-1}}$) that is too small to give a significant
haloscopic shift666For similar reasons, there is no significant haloscopic
shift of $a_{\mu}^{\pi\pi}(\tau)$, the two-pion part of $a_{\mu}$ calculated
from the semi-leptonic decay of the $\tau$ lepton:
$\tau\rightarrow\nu_{\tau}\pi^{0}\pi^{-}$. In fact,
$a_{\mu}^{\pi\pi}(\tau)=(703.0\pm 4.3)\times 10^{-10}$[4] is in good agreement
with the simulation result $a_{\mu}^{\pi\pi}({\rm Lattice})=(707.5\pm
5.5)\times 10^{-10}$ [6]. The proposed muon haloscope at JPARC[36] similarly
will see no significant haloscopic shift since the proposed four-volume is
four million times smaller than that of the Fermilab haloscope. Finally,
probes of the electron magnetic moment using a single electron in a magnetic
bottle have small four-volume and show no significant haloscopic shift[37].
However, the two-pion part $a_{e}^{\pi\pi}$ does get a haloscopic shift
$\Delta a_{e}^{\pi\pi}/a_{e}^{\pi\pi}\approx-\Delta
a_{\mu}^{\pi\pi}/a_{\mu}^{\pi\pi}$ of roughly the same size but opposite sign
due to the process $\pi^{0}+A\rightarrow\pi^{+}+\pi^{-}$. This is related to
the one in Fig. 2 by crossing symmetry: we take the $\pi^{0}$ to be incoming
and produce $\pi^{+}\pi^{-}$ as the outgoing particles..

Weighing the axion with meson haloscopy provides a quantitative check on the
consistency of our interpretation (Fig. 3). Muons give
$m_{A}g_{A\gamma\gamma}$ using Eq. (2) and pions give the axion-photon
coupling using the relation[17]:
$g_{A\gamma\gamma}f_{\pi}=\xi_{A\pi}\alpha/(2\pi)$. However, both results can
be expressed in terms of the axion decay constant $f_{A}$ by treating quark
masses as small perturbations to the quark-gluon dynamics [17, 22]:
$\xi_{A\pi}f_{A}=(0.76\pm 0.04)f_{\pi}$ and $m_{A}f_{A}=(0.459\pm
0.005)m_{\pi}f_{\pi}$. The value $f_{A}^{\mu}=(1.15\pm 0.38)\times
10^{7}~{}{\rm GeV}$ from muons and $f_{A}^{\pi}=(1.29\pm 0.42)\times
10^{7}~{}{\rm GeV}$ from pions are independent and in agreement. Combining the
muon and pion results gives the decay constant $f_{A}=(1.34\pm 0.25)\times
10^{7}~{}{\rm GeV}$ with which we can weigh the axion $m_{A}=(0.47\pm
0.11)~{}{\rm eV}$ and find its coupling to photons $g_{A\gamma\gamma}=(0.73\pm
0.17)\times 10^{-10}~{}{\rm GeV^{-1}}$. The best available astrophysical
estimate $g_{A\gamma\gamma}=(0.45\pm 0.14)\times 10^{-10}~{}{\rm GeV^{-1}}$ is
in modest tension $(1.5\sigma)$ with the haloscopic result[23].

To sharpen these results and test the haloscopic scaling laws in Eq. (1) and
(2), we make the natural assumption that axions saturate not just the local
dark matter halo of the galaxy, but also the dark matter density throughout
the observable universe. Using the baryon to photon ratio $\eta=(6.085\pm
0.090)\times 10^{-10}$, we then get the ratio of axions to photons in terms of
$m_{A}$[2, 7]:

$\displaystyle\frac{n_{A}}{n_{\gamma}}=\eta\frac{m_{B}}{m_{A}}\frac{\Omega_{A}h^{2}}{\Omega_{B}h^{2}}$
(3)

where, $m_{B}=0.938~{}{\rm GeV/c^{2}}$ is the mass of the proton, and
$\Omega_{A}h^{2}=0.1148\pm 0.0019,~{}\Omega_{B}h^{2}=0.02225\pm 0.00033$ are
the best available density parameters for dark matter and normal matter,
respectively, within the standard cosmological model[7]. To proceed, we
propose $n_{A}/n_{\gamma}=6$ as the exact value, to which we return in a
moment, and use Eq. (3) to weigh the axion giving $m_{A}=(0.492\pm
0.013)~{}{\rm eV}$. The result is consistent with the value found with meson
haloscopy but is eight times more precise. The increased precision allows a
test (see Fig. 4) of the proposed haloscopic scaling777The $\rho$ meson effect
is $\delta_{\epsilon}/2$, $N_{\rho}=1$, and we take $\rho_{A}=(0.35\pm 0.10)$
from Eq. (1). For muons and pions, we divide the observed haloscopic shifts
$\Delta a_{\mu}/a_{\mu}$ and $\Delta a_{\mu}^{\pi\pi}/a_{\mu}^{\pi\pi}$ by the
pre-factors of the square roots on the right-hand side of Eq. (2) evaluated
using $m_{A}=(0.492\pm 0.013)~{}{\rm eV}$ from Eq. (3) and
$\xi_{A\pi}=(1.66\pm 0.09)m_{A}/m_{\pi}=(6.04\pm 0.36)\times 10^{-9}$[17,
22]..

Figure 3: Weighing the Axion [Color Online] Muon (green solid) and pion (grey
dashed) haloscopy together with quark chromodynamics (blue dotted) define a
narrow acceptance window for the axion-photon coupling $g_{A\gamma\gamma}$ and
the axion mass $m_{A}$ from which we estimate (red square) $m_{A}=(0.47\pm
0.11)~{}{\rm eV}$ and $g_{A\gamma\gamma}=(0.73\pm 0.17)\times 10^{-10}~{}{\rm
GeV^{-1}}$.

Before discussing current and future meson haloscope experiments, we come back
to the physics at $f_{A}\approx 10^{7}~{}{\rm GeV}$ and the origin of the
exact value $n_{A}/n_{\gamma}=6$ that we used to weigh the axion precisely and
to test haloscopic scaling. It is straightforward to check that the axion
transforms in the same way as the following fermion bilinear888The axion
$A_{Q}^{P}$ transforms as $CA_{Q}^{P}=-PA_{Q}^{P}=A_{Q}^{P}$ under charge
conjugation $C$ and parity $P$ symmetries. Similarly, the photon $\gamma_{H}$
transforms as $C\gamma_{H}=\gamma_{H},~{}P\gamma_{H}=\gamma_{\overline{H}}$.
These fix the form of the expressions for the axion and photon in terms of
chiral fermions $M_{H}^{P}$ which transform as
$CM_{H}^{P}=\overline{M}_{H}^{\overline{P}},~{}PM_{H}^{P}=M_{\overline{H}}^{P}$
with $M=Q\cup L$:
$A_{Q}^{P}=i(\sum_{G}Q_{L}^{P}\overline{Q}_{R}^{\overline{P}}-Q_{R}^{P}\overline{Q}_{L}^{\overline{P}})/\sqrt{6}$.
Similarly, the photon transforms in the same way as
$\gamma_{H}=\sum_{G,M,P}M_{H}^{P}\overline{M}_{H}^{\overline{P}}/(4\sqrt{3})$.
Here, the chiral fermions $M_{H}^{P}$ have helicity $H=\\{L,R\\}$ and charge
parity $P=\\{0,1\\}$, and $M=Q\cup L$ are the quarks
$Q=\\{U^{r},D^{r},U^{g},D^{g},U^{b},D^{b}\\}$ and leptons $L=\\{e,\nu_{e}\\}$
of the first generation of the standard model while the sums on $G$ run over
all three generations. Remarkably, these expression gives the exact value
$n_{A}/n_{\gamma}=|Q||P|/|H|=6$ suggested by meson haloscopy, provided that
axions of each flavor $A_{Q}^{P}$ and photons of each helicity $\gamma_{H}$
are made in equal numbers.

In the standard model, charge parity and helicity are related[1]: we have
chiral fermions $M_{L}^{1}$ and $\overline{M}_{R}^{0}$, but neither
$M_{L}^{0}$ nor $\overline{M}_{R}^{1}$ are found. If we assume that this
correlation emerges only at the energy scale $10^{7}~{}{\rm GeV}$, then, above
$f_{A}$, the number of chiral fermions doubles. The strength of the nuclear
forces and electromagnetism change with energy scale in a way that depends
sensitively on the number of chiral fermions[1]. Doubling the number of chiral
fermions above $10^{7}~{}{\rm GeV}$ changes the energy scale where the
strengths of the nuclear forces become equal, for example. Although we defer
detailed investigation to a later work, it seems likely that the physics at
$f_{A}\approx 10^{7}~{}{\rm GeV}$ is relevant not just for setting the value
$n_{A}/n_{\gamma}=6$, but also for the prospects of unifying the nuclear
forces and electromagnetism.

The current tension between muon and pion haloscopy $\Delta a_{\mu}+|\Delta
a_{\mu}^{\pi\pi}|=(25.1\pm 5.9)\times 10^{-10}$ sits at 4.2$\sigma$, just shy
of the 5$\sigma$ threshold for discovery of a new phenomenon[5, 4]. To cross
the threshold at the current level of precision, it is enough to boost one of
the signals by 40 per cent. One way to do that is to cut the tune $n\approx
0.11$ of the muon haloscope in half while holding constant the pitch $C_{p}$
and field $C_{e}$ corrections[26]: the muon haloscopic shift scales as $\Delta
a_{\mu}\propto\sqrt{V_{A\mu}}\propto 1/\sqrt{n}$, for $n$ much less than one
at constant $C_{p}$ and $C_{e}$. This change can be made in the next run of
the Fermilab muon haloscope by cutting in half the voltage on the four
electrostatic quadrupole plates which focus the muon beam[26].

Figure 4:  Meson Haloscopy Scaling [Color Online] The observed meson
haloscopic effects scale with haloscope 4-volume as
$\sqrt{V_{AX}N_{X}/\Gamma_{X}}$ (purple line) for $X=\mu~{}({\rm
green~{}square}),\pi~{}({\rm blue~{}triangle}),\rho~{}({\rm orange~{}circle})$
mesons. There are no adjustable parameters. The uncertainty in the line is
comparable to its thickness while those of the points are similiar to their
size.

The inverse pion haloscopic effect (Fig. 1, with $\pi^{0}$ viewed as an
incoming particle) can be seen with current pion haloscopes by tuning to the
node in $\pi^{+}\pi^{-}$ production at energy $\sqrt{s}=1.6~{}{\rm GeV}$[12].
The node coincides with the $\rho^{\prime\prime}(1700)$ resonance for
$\pi^{+}\pi^{-}\pi^{0}$ production[27]. A pion haloscope with large four-
volume, such as the KLOE detector at the DAFNE collider, will see an excess of
2$\pi$ events and a deficit of 3$\pi$ events at this energy compared to
haloscopes with small four-volume, such as, the BABAR detector at the PEP-II
collider[27].

The tensions between probes of the muon magnetic moment are due to axions
which saturate the local dark matter halo of the galaxy. We resolve the
tension by weighing the axion using meson haloscopy. Increasing the muon
haloscopic volume can increase the tension beyond the threshold for discovery
during the next run at Fermilab. A pair of haloscopes with different volume,
such as BABAR and KLOE, can test the haloscopic principle directly. A symmetry
argument suggests that there are six axions per photon in the observable
universe and that the number of chiral fermions doubles above $10^{7}~{}{\rm
GeV}$.

## Acknowledgements

We thank U.-J. Wiese, M. Hoferichter, G. Colangelo, D. Stockinger, and D.
Hertzog for insightful correspondence.

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  * Note [4] We take the muon haloscope parameters from Fermilab’s E989 experiment[5] and the pion parameters from the KLOE detector at the DAFNE collider[31]. The volume of the muon haloscope is [5, 26, 33] $V_{A\mu}=2\pi^{2}R^{3}\sqrt{C_{e}C_{p}}/(n\sqrt{1-n})=(6.29\pm 0.42)\times 10^{3}~{}{\rm cm^{3}}$, where, $R=711~{}{\rm cm}$ is the radius of the central orbit of muons in the ring, $C_{e}$ is the electric field correction from the radial width of the muon beam, $C_{p}$ is the pitch correction from the width of the beam along the magnetic field, $n\approx 4\times(2\omega_{c}-\omega_{\rm BO})/(2\omega_{c})\approx 0.11$ is the index for the electrostatic quadrupole field, $\omega_{c}/(2\pi)=c/(2\pi R)=6.72~{}{\rm MHz}$ is the cyclotron frequency, and $\omega_{\rm BO}/(2\pi)=13.1~{}{\rm MHz}$ is the betatron oscillation frequency. The number of muons at the start of haloscopy is [5, 26]: $N_{\mu}=N_{0}e^{-\Gamma_{\mu}t_{\rm fill}}\approx 3100$, where, $N_{0}\approx 5000$ is the number of muons in the ring at the start of the fill, and $t_{\rm fill}\approx 30~{}\mu{\rm sec}$ is the time it takes to fill the ring. The muon decay rate in the haloscope rest frame is $\Gamma_{\mu}=\hbar/(\gamma_{\mu}\tau_{\mu})=1.20\times 10^{-20}~{}{\rm GeV}$, where, $\gamma_{\mu}\approx p_{\mu}/(m_{\mu}c)=29.3$ is the muon time dilation factor, $p_{\mu}=3.10~{}{\rm GeV/c}$ is the muon momentum, $m_{\mu}=0.106~{}{\rm GeV/c^{2}}$ is the muon rest mass, and $\tau_{\mu}=2.20~{}\mu{\rm sec}$ is the muon life-time in its rest frame[5]. The volume of the pion haloscope is:$V_{A\pi}=V_{A\rho}=1.2\times 10^{-3}~{}{\rm cm^{3}}$. The number of pions at the start of haloscopy is [34, 35]: $N_{\pi}=2(a_{\mu}^{\pi\pi}/\sigma^{\rm stat}_{a_{\mu}^{\pi\pi}})^{2}=1.23\times 10^{5}$, where, $\sigma^{\rm stat}_{a_{\mu}^{\pi\pi}}=1.1\times 10^{-10}$ is the statistical uncertainty of $a_{\mu}^{\pi\pi}=385.1\times 10^{-10}$. The haloscope bandwidth is set by the light-crossing time of the luminous region[31]:$\Gamma_{\pi}=\hbar c/\sigma_{\rm min}=9.87\times 10^{-12}~{}{\rm GeV}$, where, $\sigma_{\rm min}={\rm min}\left\\{\sigma_{x},\sigma_{y},\sigma_{z}\right\\}=\sigma_{y}$ for the KLOE detector at DAFNE.
  * Note [5] The haloscopic shift of the muon magnetic moment is $\Delta a_{\mu}=a_{\mu}({\rm Exp})-a_{\mu}({\rm SM})+a_{\mu}^{\pi\pi}({\rm SM})-a_{\mu}^{\pi\pi}({\rm Lattice})=(10.6\pm 6.7)\times 10^{-10}$, where, $a_{\mu}({\rm Exp})-a_{\mu}({\rm SM})=25.1\pm 5.9)\times 10^{-10}$ is the shift of the world average muon measurement $a_{\mu}({\rm Exp})$[5] from the Standard Model value $a_{\mu}({\rm SM})$[4]. Here, $a_{\mu}^{\pi\pi}({\rm SM})=(693.1\pm 4.0)\times 10^{-10}$ is the measured two-pion part of the muon magnetic moment[4, 12, 11, 10] and $a_{\mu}^{\pi\pi}({\rm Lattice})=(707.5\pm 5.5)\times 10^{-10}$ is the simulation result found using quark chromodynamics[6]. Similarly, the haloscopic shift of the two-pion part is $\Delta a_{\mu}^{\pi\pi}=a_{\mu}^{\pi\pi}({\rm Exp})-a_{\mu}^{\pi\pi}({\rm Lattice})=-(18.7\pm 5.8)\times 10^{-10}$, where, we use the KLOE detector at the DAFNE collider experimental result[10, 4] for $a_{\mu}^{\pi\pi}({\rm Exp})=(688.8\pm 1.8)\times 10^{-10}$.
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