# The radiation emitted from axion dark matter in a homogeneous magnetic
field, and possibilities for detection

Shuo Xu  School of Physics, Sun Yat-sen University, Guangzhou, GuangDong,
People’s Republic of China  Department of Astronomy, Tsinghua University,
People’s Republic of China  Siyu Chen  Hong-Hao Zhang<EMAIL_ADDRESS>Guangbo Long<EMAIL_ADDRESS>School of Physics, Sun Yat-sen
University, Guangzhou, GuangDong, People’s Republic of China

###### Abstract

We study the direct radiation excited by oscillating axion (or axion-like
particle) dark matter in a homogenous magnetic field and its detection scheme.
We concretely derive the analytical expression of the axion-induced radiated
power for a cylindrical uniform magnetic field. In the long wave limit, the
radiation power is proportional to the square of the B-field volume and the
axion mass $m_{a}$, whereas it oscillate as approaching the short wave limit
and the peak powers are proportional to the side area of the cylindrical
magnetic field and $m_{a}^{-2}$. The maximum power locates at mass
$m_{a}\sim\frac{3\pi}{4R}$ for fixed radius $R$. Based on this characteristic
of the power, we discuss a scheme to detect the axions in the mass range
$1-10^{4}$ neV, where four detectors of different bandwidths surround the
B-field. The expected sensitivity for $m_{a}\lesssim 1\,\mu$eV under typical-
parameter values can far exceed the existing constraints.

##  I Introduction

The axion originates from the Peccei-Quinn solution to the strong $CP$
problem. Its generalization, axionlike particles (ALPs, in the following
referred to as axion), are predicted by many extensions of the Standard Model
Svrcek:2006yi . These well-motivated pseudoscalar particles of electrically-
neutral are also the prime candidates for the dark matter (DM) Preskill1983 ;
Abbott1983 ; Dine1983 .

The interaction between the axion and the photon is described by the
Lagrangian ${\cal L}_{a\gamma}=g_{a\gamma}{\bf E}\cdot{\bf B}\,a$, where ${\bf
E}$ represents the electric field, ${\bf B}$ the magnetic field, $a$ the
axion, and $g_{a\gamma}$ the coupling constant. If the axion is the DM
particle, there is a oscillating axion field $a(t)$ in our Galactic halo. The
external magnetic field ${\bf B}_{\rm e}$ with the axion field induces an
extra source term in Maxwell’s equations as the effective electric current
density Millar2017JCAP

${\bf J}_{a}(t)=g_{a\gamma}{\bf B}_{\rm e}\dot{a}(t).$ (1)

Therefore, the energy of axion dark mater (ADM) can be converted to the
electromagnetic field and the converted photons is expected to be observed as
long as the B-field is strong enough. A number of experimental schemes have
been designed to detect this axion-induced signal Sikivie:2020zpn .

At present, most of the existing or proposed experiments (haloscopes) take the
advantage of the resonant enhancement on the EM modes drived by ${\bf
J}_{a}(t)$ Sikivie:2020zpn . For $1\,\mu{\rm eV}\lesssim m_{a}\lesssim
50\,\mu$eV, The cavity haloscopes (e.g. ADMX ADMX:2021nhd , WISPDMX WISPDMX ,
CAPP CAPP:2020utb , HAYSTAC Brubaker:2016ktl , QUAX Alesini:2019ajt , GrAHal
Grenet:2021vbb , RADES CAST:2020rlf ,ORGAN Goryachev:2017wpw ) in strong
magnetic fields are optimal strategy, where a narrow-band axion-induced
excitation around each of the cavity resonances can be achieved Sikivie:1983ip
. For smaller mass $m_{a}\lesssim 1\,\mu$eV, it can be explored by LC circuit
DMRadio ; Cabrera2008 ; Sikivie:2013laa ; Kahn:2016aff ; Zhang:2021bpa ;
Chen:2021bgy  or radio-frequency cavity haloscopes Berlin:2019ahk ;
Bogorad:2019pbu ; Lasenby:2019prg ; Berlin:2020vrk ; Sikivie1009 . The former
has been experimentally realized by SHAFT Gramolin:2020ict , ABRACADABRA
Ouellet:2018beu , ADMX-SLIC Crisosto:2019fcj  and the BASE Penning trap
Devlin:2021fpq . For higher mass $50\,\mu{\rm{eV}}\lesssim
m_{a}\lesssim\,1\,$eV, the traditional experiment of resonance cavity is
limited by the smaller volume required by the resonant enhancement and
impractical scan rates for high mass. As a result, several novel detection
techniques of resonance or constructive interference, such as topological
insulators target Marsh:2018dlj ; Schutte-Engel:2021bqm , plasma Haloscopes
Lawson:2019brd , multiple-cavity Aja:2022csb ; Jeong:2020mtp ; Melcon:2018dba
; Baryakhtar:2018doz  and dielectric haloscopes Caldwell:2016dcw ;
MADMAX:2019pub ; Millar2017JCAP , have been proposed to exploit the ADM of
this mass band.

Another type of strategy to probe the high-mass ADM is the dish-antenna
Horns:2012jf  or solenoidal haloscope BREAD:2021tpx  without resorting to the
resonant enhancement on the signal. The designed detector catch the axion-
induced emission from the magnetized dish-antenna or cylindrical metal barrel
surface and the detectable power is proportional to the area of the reflecting
surface $A_{\rm ref}$ Horns:2012jf ; Jaeckel:2013eha . These haloscopes
compensate the loss of the resonance enhancement by increasing the volume of
the B-field and can work in a broadband state. Note that, the dielectric
haloscopes Caldwell:2016dcw ; MADMAX:2019pub ; Millar2017JCAP  evolved from
the dish-antenna scheme.

In this work, we generalize the strategy of dish-antenna to a fully open case
without a reflector in the B-field, which (or a similar case) was also
mentioned in an preprint Horns:2013ira . We focus on investigating the direct
radiation excited by the oscillating electric current density ${\bf
J}_{a}(t)$, and derive the analytical expression of radiated power for a
cylindrical uniform B-field concretely, which is more common in scientific
detection instruments. Then, we discuss the possibility of detecting the ADM
by means of the derived radiated power.

The plan of this paper is as follows. In Sec. II, we will derive the radiated
power and the energy flux from axion conversion in a homogeneous magnetic
field. Then, in Sec. III we will discuss the experimental detection of the
ADM. Finally, discussions and conclusions are presented in Sec. IV and V
respectively.

##  II The radiated power from axion conversion

This section derives the radiated power and the energy flux from the axion-
photon conversion in a homogeneous static B-field $\mathbf{B}_{e}$. We use the
Heaviside-Lorentz units, in which $c=1$, $k_{\rm B}=1$, the permeability and
permitivity of the vacuum $\mu_{0}=\epsilon_{0}=1$.

###  II.1 Axion electrodynamics

We assume the axion field
$a(t)={\rm{Re}}(a_{0}\,e^{\rm{i}\,(\mathbf{k}_{a}\cdot\mathbf{x}-\omega t)})$
$\simeq$ ${\rm{Re}}(a_{0}\,e^{-\rm{i}\,\omega t})$ since ADM particles move
non-relativistically based on $k_{a}\ll\omega$ and $\omega=m_{a}$. The
retarded potential can be obtained from solving Maxwell’s equations with the
only source term $\mathbf{J}_{a}$ in Eq. (1) Sikivie:2020zpn

$\mathbf{A}(\mathbf{x})={1\over 4\pi}\int_{V}{\mathbf{j}_{a}\,{\rm e}^{-{\rm
i}\,kr}\over r}~{}dV^{\,\prime},$ (2)

where $k=\omega$, $\mathbf{j}_{a}={\rm i}\,g_{a\gamma}\omega
a_{0}\mathbf{B}_{\rm e}$, and $\epsilon=\mu=1$. Here,
$r=|\mathbf{x}-\mathbf{x}^{\,\prime}|>$ 0 is the distance from the observation
location $\mathbf{x}$ to $\mathbf{x}^{\,\prime}$ of the magnetic field region.
The range of integration is in the region extended by the B-field volume $V$.

The axion-induced radiated the energy-flux $\mathbf{S}$ for E-field
$\mathbf{E}$ and B-field $\mathbf{B}$ is given by

$\langle\mathbf{S}\rangle\,=\frac{\rm
Re(\mathbf{E}^{*}\times\mathbf{B})}{2},\,\,\,\mathbf{B}=\bigtriangledown\times\mathbf{A},\,\,\,\mathbf{E}=\frac{\rm
i}{\omega}\bigtriangledown\times\mathbf{B}.$ (3)

The bracket $\langle...\rangle$ represents a time average.

For convenience, we calculate the radiated power from the axion conversion in
the Fraunhofer radiated zone, i.e., $r>\frac{L^{2}\omega}{\pi}$ and
$r\simeq|\mathbf{x}|\gg L$, where $L$ is the feature size of the B-field
region. The retarded potential in Eq. (2) can be reduced to

$\mathbf{A}(\mathbf{x})={{\rm e}^{{\rm i}\,kr}\over 4\pi
r}\,\mathbf{j}_{a}(\mathbf{k})+{\rm O}({1\over r^{2}}),$ (4)

where $\mathbf{j}_{a}(\mathbf{k})=\int_{V}{{\rm e}^{-{\rm
i}\,\mathbf{k}\cdot\,\mathbf{x}^{\,\prime}}\mathbf{j}_{a}}~{}dV^{\,\prime}$ is
the Fourier transform of $\mathbf{j}_{a}$. The direction of the wave vector
satisfies $\mathbf{n}=\mathbf{k}/k=\mathbf{r}/r$.

The averaged electromagnetic power $dP$ radiated for area d$\mathbf{s}$ in
direction $\mathbf{n}$ can be represented by $dP=\langle\mathbf{S}\rangle\cdot
d\mathbf{s}$=$\langle\mathbf{S}\rangle\cdot\mathbf{n}\,r^{2}d\Omega$. Then,
the time-averaged power in the Fraunhofer radiated zone can be estimated as

$\displaystyle P$ $\displaystyle=$
$\displaystyle\int_{\Omega}\langle\mathbf{S}\rangle\cdot\mathbf{n}\,r^{2}\,d\Omega$
(5) $\displaystyle=$ $\displaystyle\int_{\Omega}{k\omega\over
32\pi^{2}}|\mathbf{n}\times\mathbf{j}_{a}(\mathbf{k})|^{2}\,d\Omega$
$\displaystyle=$
$\displaystyle\frac{g_{a\gamma}k\omega^{3}|a_{0}|^{2}}{32\pi^{2}}\,\int_{\Omega}\,\bigg{|}\int_{V}\,\mathbf{n}\times\mathbf{B_{\rm
e}}\,{{\rm e}^{-{\rm
i}\,\mathbf{k}\cdot\,\mathbf{x}^{\,\prime}}}dV^{\,\prime}\bigg{|}^{2}\,d\Omega$
$\displaystyle=$ $\displaystyle{\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}\omega^{2}\over
16\pi^{2}}\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta\,{\rm
sin}^{3}\theta\,\bigg{|}\int_{V}\,{{\rm e}^{-{\rm
i}\,\mathbf{k}\cdot\,\mathbf{x}^{\,\prime}}}dV^{\,\prime}\bigg{|}^{2}$

where $\rho_{a}=\frac{\omega^{2}|a_{0}|^{2}}{2}$ is the energy density of the
ADM, and we asumme $|\mathbf{n}\times\mathbf{B_{\rm e}}|=B_{\rm e}\,{\rm
sin}\theta$ for the homogenous extra B-field (see, Fig. 1).

Note that the excited E-field in the homogeneous B-field is also an
oscillating field ${\bf E}_{a}(t)=-g_{a\gamma}{\bf B}_{\rm e}\,a(t)$
Millar2017JCAP .

###  II.2 The axion-induced radiated power for a cylindrical shape of the
magnetic field

Figure 1: The schematic representation of a cylindrical magnetic field. The
origin of x-y-z coordinate is located at the center of the cylindrical
magnetic field which is parallel to the z axis. The vector $\mathbf{n}$ shows
the direction of the wave vector of the radiated electromagnetic waves
($\gamma$) from axion ($a$) conversion, and its angle to the $z$ axis is
denoted by $\theta$. The height and the radius of the cylindrical B-field is
denoted by $h$ and $R$, respectively.

We study the axion-induced radiation in a cylindrical uniform B-field with
height $h$ and radius $R$, as shown in Fig. 1. The time-averaged radiated
power can be further simplified by integrating over the volume of the B-field
in Eq. (5). We calculate $\int_{V}\,{{\rm
e}^{-i\,\mathbf{k}\cdot\,\mathbf{x}^{\,\prime}}}dV^{\,\prime}$ in cylindrical
coordinate for $\mathbf{x}^{\,\prime}$ and in spherical coordinate for
$\mathbf{k}$ as

$\displaystyle I_{V}$ $\displaystyle=$
$\displaystyle\int_{-\frac{h}{2}}^{\frac{h}{2}}{\rm exp}(-{\rm i}k{\rm
cos}\theta z)dz\,\int_{0}^{2\pi}d\phi\int_{0}^{R}d\rho\,{\rm exp}(-{\rm i}\rho
k{\rm sin}\theta$ (6) $\displaystyle({\rm cos}\varphi{\rm cos}\phi+{\rm
sin}\varphi{\rm sin}\phi))$ $\displaystyle=$ $\displaystyle\frac{2{\rm
sin}(\frac{h}{2}k{\rm cos}\theta)}{k{\rm
cos}\theta}\int_{0}^{R}d\rho\int_{0}^{2\pi}d\phi\,{\rm e}^{-{\rm i}\rho k{\rm
sin}\theta{\rm cos}(\varphi-\phi)}$ $\displaystyle=$ $\displaystyle\frac{2{\rm
sin}(\frac{h}{2}k{\rm cos}\theta)}{k{\rm
cos}\theta}\,\int_{0}^{R}d\rho\int_{0}^{2\pi}d\phi\,{\rm e}^{-{\rm i}\rho
k{\rm sin}\theta{\rm cos}(\phi)},$

where we have used the column symmetry of the radiation and have chosen
$\varphi$=0. By means of the zero-order Bessel-function ${\rm
J}_{0}(x)=\frac{1}{2\pi}\int_{0}^{2\pi}{\rm exp}(-{\rm i}x{\rm
cos}(\phi))d\phi$, the representation of $I_{V}$ can be reduced to

$\displaystyle I_{V}$ $\displaystyle=$ $\displaystyle\frac{2{\rm
sin}(\frac{h}{2}k{\rm cos}\theta)}{k{\rm cos}\theta}\,\int_{0}^{R}2\pi{\rm
J}_{0}(\rho k{\rm sin}\theta)\rho\,d\rho$ (7) $\displaystyle=$
$\displaystyle\frac{2{\rm sin}(\frac{h}{2}k{\rm cos}\theta)}{k{\rm
cos}\theta}\frac{2\pi}{(k{\rm sin}\theta)^{2}}\int_{0}^{Rk{\rm
sin}\theta}(\rho k{\rm sin}\theta){\rm J}_{0}(\rho k{\rm sin}\theta)$
$\displaystyle d(k\rho{\rm sin}\theta)$ $\displaystyle=$
$\displaystyle\frac{2{\rm sin}(\frac{h}{2}k{\rm cos}\theta)}{k{\rm
cos}\theta}\,2\pi R^{2}\frac{{\rm J}_{1}(Rk{\rm sin}\theta)}{Rk{\rm
sin}\theta},$

where we have used the relation of $\int x{\rm J}_{0}(x)=x{\rm J}_{1}(x)$ in
the last step of the derivation. The integral over $\rho$ and $\phi$ above
implies the Fraunhofer diffraction of a circular hole. Therefore, one can
obtain the time-averaged radiated power for the “cylindrical” B-field as

$\displaystyle P$ $\displaystyle=$
$\displaystyle{\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}\omega^{2}\over
16\pi^{2}}\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta\,{\rm
sin}^{3}\theta\,|I_{V}|^{2}$ (8) $\displaystyle=$
$\displaystyle{2\pi\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}R^{4}}\int_{0}^{\pi}{\rm sin}^{3}\,\theta\Big{(}\frac{{\rm
sin}(\frac{h}{2}\omega{\rm cos}\theta)}{{\rm cos}\theta}\Big{)}^{2}\,$
$\displaystyle\Big{(}\frac{{\rm J}_{1}(R\omega{\rm sin}\theta)}{R\omega{\rm
sin}\theta}\Big{)}^{2}d\theta.$

Figure 2: The integral functions ($f(\theta)$) in Eq. (8) for different sizes
of the B-field volume, where the yellow lines represent the approximated
results with taking $\theta=\pi/2$ in the aperture diffraction function
$\big{(}\frac{{\rm J}_{1}(R\omega{\rm sin}\theta)}{R\omega{\rm
sin}\theta}\big{)}^{2}$. Top panel: $h=3\frac{2\pi}{\omega}\approx 3.7\,$m,
$R=\frac{\pi}{\omega}\approx 0.6\,$m, and the ordinate is shown in logarithmic
form; Middle panel: $h=\frac{2\pi}{\omega},R=3\frac{\pi}{\omega}$, and the
ordinate is shown in logarithmic form; Bottom panel:
$h=\frac{2\pi}{\omega}=2R$.

Now, the integral above cannot be further calculated analytically and we need
to study the integral function to approximate it. Fig. 2 shows the numerical
plots of the integral function in Eq. (8) for different sizes of the B-field
volume, where the yellow lines represent a approximation with taking
$\theta=\pi/2$ in the “aperture diffraction function” $\big{(}\frac{{\rm
J}_{1}(R\omega{\rm sin}\theta)}{R\omega{\rm sin}\theta}\big{)}^{2}$. The
ordinates of the top and middle panels are shown in logarithmic form. We can
find that the radiation is concentrated around the horizontal direction
($\theta=\frac{\pi}{2}$), which is similar to the principal maximum of
Fraunhofer diffraction. The secondary peak number in the radiation angular
distribution is $N=2(N_{h}+N_{R}-2)$ when $N_{h}=\frac{h\omega}{2\pi}$ and
$N_{R}=\frac{R\omega}{\pi}$ are integers. When $h>\frac{2\pi}{\omega}$ and
$R>\frac{\pi}{\omega}$, the radiation is more concentrated horizontally. The
approximations are more accurate for $h\gtrsim 2R$, which means the dominance
of the longitudinal radiation over the overall radiation, as shown in the top
panel.

Inspired from the above analysis, we can use the delta function to approximate
the longitudinal integral function as

$\displaystyle P$ $\displaystyle=$
$\displaystyle{2\pi\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}R^{4}}\int_{0}^{1}-{\rm
sin}^{3}\theta\pi\frac{{\rm sin^{2}}(\frac{\omega h}{2}{\rm
cos}\theta)}{\pi\frac{\omega h}{2}{\rm cos}^{2}\theta}\frac{\omega h}{2}$ (9)
$\displaystyle\Big{(}\frac{{\rm J}_{1}(R\omega{\rm sin}\theta)}{R\omega{\rm
sin}\theta}\Big{)}^{2}\frac{d({\rm cos}\theta)}{{\rm sin}\,\theta}$
$\displaystyle\simeq$ $\displaystyle{2\pi^{2}\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}R^{4}}\int_{0}^{1}-{\rm sin}^{3}\,\theta\delta({\rm
cos}\theta)\frac{\omega h}{2}$ $\displaystyle\Big{(}\frac{{\rm
J}_{1}(R\omega{\rm sin}\theta)}{R\omega{\rm sin}\theta}\Big{)}^{2}\frac{d({\rm
cos}\theta)}{{\rm sin}\,\theta}$ $\displaystyle=$
$\displaystyle{\pi^{2}\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}R^{4}h\omega}\Big{(}\frac{{\rm J}_{1}(R\omega)}{R\omega}\Big{)}^{2}.$

To ensure the validity of the $\delta$ approximation, $\frac{\omega h}{2}$ is
required to be sufficiently large compared to the other variate or parameter
of the internal function, such as $\rm cos\theta$ and $\omega R$. Here, we
require $\frac{\omega h}{2}>1$ (the maximum of ${\rm cos}\theta$) to ensure
the horizontal focusing of the radiation and $h\gtrsim 2R$ to ensure the
dominance of the longitudinal radiation, as analyzed above.

We further expand the Bessel function for $\omega R>1$ so that Eq. (9) is
reduced to

$\displaystyle P$ $\displaystyle\simeq$ $\displaystyle
2\pi\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}\frac{Rh}{\omega^{2}}{\rm
cos}^{2}(R\omega-\frac{3\pi}{4}),\,\,(\frac{\omega h}{2}\gtrsim\omega R>1).$

We are more interested in the extremum (resonance point) of the power $P$,
around which the $\delta$ approximation is also more valid. The power at the
extreme points is given by

$\displaystyle P$ $\displaystyle=$ $\displaystyle\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}\frac{A}{\omega^{2}},\,\big{(}\frac{\omega h}{2}\gtrsim
R\omega=(n+\frac{3}{4})\pi,n=0,1,2\cdot\cdot\cdot\big{)}$ (11)
$\displaystyle=$ $\displaystyle 6.9\times 10^{-23}{\rm
W}\,\frac{\rho_{a}}{0.3\,\rm GeV\,cm^{-3}}\frac{h}{2\,\rm m}\frac{R}{\rm
m}\Big{(}\frac{B_{\rm e}}{10\,\rm T}\Big{)}^{2}$
$\displaystyle\Big{(}\frac{\rm\mu
eV}{m_{a}}\Big{)}^{2}\Big{(}\frac{g_{a\gamma}}{10^{-14}\rm
GeV^{-1}}\Big{)}^{2},\,\,\,\big{(}h\gtrsim 2R,$ $\displaystyle 5.1\frac{R}{\rm
m}\frac{m_{a}}{\mu\rm
eV}=(n+\frac{3}{4})\pi,\,n=0,1,2\cdot\cdot\cdot\big{)}\,\,\,,$

where $A=2\pi Rh$ is the side area of the cylinder. The extreme radiated power
is proportional to the surface area of B-field region and is inversely
proportional to the square of the frequency, which is the same as the disk
antenna experiment Horns:2012jf .

Figure 3: The comparison between the radiated power (blue line) resulted from
numerical integral according to Eq. (8) and its approximation (yellow line)
according to Eq. (LABEL:pJ). Top panel: P as a function of the radius $R$ for
$h=2R$ and $\omega=1\mu\,$eV; Middle panel: Like the case of the top panel,
but $h$ is fixed to 2 m; Bottom panel: P as a function of the frequency for
$h=2\,$m and $R=1\,$m. The approximation is good around the peaks when
$\frac{\omega h}{2}\gtrsim\omega R>1$. When $\omega R<1$, the approximation
fails as shown in all figures and also for the case $h<2R$ in the middle
panel. The other parameter values are chosen as $g_{a\gamma}=10^{-12}\,{\rm
GeV^{-1}}$, $B=10\,{\rm T}$, and $\rho_{a}=0.3\,{\rm GeVcm^{-3}}$. Figure 4:
The radiated power at extreme points varies with R according to Eq. (11). The
other parameter values are taken as in Fig. 3.

Fig. 3 shows the comparison between the radiated power (blue line) resulting
from the numerical integral according to Eq. (8) and its approximation (yellow
line) according to Eq. (LABEL:pJ). The approximation is good around the peaks
when $\frac{\omega h}{2}\gtrsim\omega R>1$. However, when $\omega R<1$, the
approximation fails at small $R$ or $\omega$ as shown in all three figures,
and the middle panel shows a similar case of $h<2R$ when $R>2\,$m.

For a fixed B-field volume, the maximum of the radiated power is the first
peak at $\omega=\frac{3}{4R}\pi$ ($R\omega=(n+\frac{3}{4})\pi>1$, n=0), before
which the power $P(\omega)$ is an increasing function of the frequency
$\omega^{2}$, and after which decreasing as $\frac{1}{\omega^{2}}$. This
feature helps us find the best measurement frequency for a fixed B-field
volume.

Fig. 4 shows the radiated power of Eq. (11) at extreme points which varies
with R. The spacing between the extreme points is very small, which means that
detecting such peak powers for $m_{a}=100\,\mu$eV requires the centimeter-
level accuracy of the magnetic field device.

The results are obtained in the short wave approximation for the radiated
power. The result of the long wave approximation is easy to obtain under the
approximation of ${{\rm e}^{-{\rm
i}\,\mathbf{k}\cdot\,\mathbf{x}^{\,\prime}}}\simeq 1$, that is

$\displaystyle P$ $\displaystyle=$
$\displaystyle{\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}\omega^{2}\over
16\pi^{2}}\int_{0}^{2\pi}d\varphi\,\int_{0}^{\pi}d\theta\,{\rm
sin}^{3}\theta\,\bigg{|}\int_{V}\,dV^{\,\prime}\bigg{|}^{2},$ (12)
$\displaystyle=$ $\displaystyle{\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}V^{2}\omega^{2}\over 6\pi},~{}\ (R\omega\ll 1,\,h\omega\ll 1)$
$\displaystyle=$ $\displaystyle 7.8\times 10^{-23}{\rm
W}\,\frac{\rho_{a}}{0.3\rm GeV\,cm^{-3}}\Big{(}\frac{B_{\rm e}}{10\rm
T}\Big{)}^{2}\Big{(}\frac{m_{a}}{\rm 100\,neV}\Big{)}^{2}$
$\displaystyle\Big{(}\frac{g_{a\gamma}}{10^{-14}\rm
GeV^{-1}}\Big{)}^{2}\Big{(}\frac{h}{\rm 2m}\Big{)}^{2}\big{(}\frac{R}{\rm
m}\big{)}^{4},\,\,\,(0.51\frac{R}{\rm m}\frac{m_{a}}{100\rm neV}$
$\displaystyle\ll 1,\,0.51\frac{h}{\rm m}\frac{m_{a}}{100\rm neV}\ll 1).$

The radiated power is proportional to the square of the B-field volume and the
frequency, and is independent of the shape of the homogenous B-field region.
Fig. 5 (yellow line) shows the power in Eq. (12) as a function of $R$ and
$\omega$, respectively. Blue lines represent the radiated power (blue line)
according to Eq. (8). We can find the approximation is consistent with the
numerical results when the condition $\omega R\ll 1$ is satisfied.

Figure 5: The comparison between the radiated power (blue line) according to
Eq. (8) and its approximation in the long wave limit (yellow line) as Eq.
(12). Top panel: $P$ as a function of the radius for $\omega=1\mu\,$eV; Bottom
panel: $P$ as a function of the frequency for $h=2\,$m and $R=1\,$m. The
approximation is consistent with the numerical results when the condition
$\omega R\ll 1$ is satisfied. The other parameter values are chosen as
$g_{a\gamma}=10^{-12}\,{\rm GeV^{-1}}$, $B=10\,{\rm T}$, and
$\rho_{a}=0.3\,{\rm GeVcm^{-3}}$.

This calculation pattern for the radiated power under the homogenous
“cylindrical magnetic field” above can be generalized to the case with a
B-field of cubic volume.

###  II.3 The energy flux

The radiated power is independent of the distance to the radiation source, but
the energy flux depends. In this part, we give the formula of the energy flux
, which may be used to detect the axion-induced radiation.

The electric and magnetic fields can be derived with Eq. (2) and Eq. (3) as

$\displaystyle\mathbf{B}$ $\displaystyle=$ $\displaystyle-{\rm
i}\frac{a_{0}\omega
g_{a\gamma}B_{e}}{4\pi}\left(I_{1y}\mathbf{e}_{x}-I_{1x}\mathbf{e}_{y}\right)$
(13) $\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle\frac{a_{0}g_{a\gamma}B_{e}}{4\pi}\big{(}I_{2xz}\mathbf{e}_{x}+I_{2yz}-(I_{2x^{2}}+I_{2y^{2}}+2I_{0})\mathbf{e}_{y}\big{)}$

with x-y-z coordinate basis vector $\mathbf{e}_{x}$, $\mathbf{e}_{y}$, and
$\mathbf{e}_{z}$. The integrals $I$ are defined as

$\displaystyle I_{0}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}\frac{{\rm i}\omega r-1}{r^{3}}e^{{\rm i}\omega
r}dV^{\prime}$ (15) $\displaystyle I_{1x}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}\frac{{\rm i}\omega r-1}{r^{3}}e^{{\rm i}\omega
r}(x-x^{\prime})dV^{\prime}$ (16) $\displaystyle I_{2x^{2}}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}\frac{3-3i\omega
r-\omega^{2}r^{2}}{r^{5}}e^{{\rm i}\omega r}(x-x^{\prime})^{2}dV^{\prime}$
(17) $\displaystyle I_{2xz}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}\frac{3-3i\omega
r-\omega^{2}r^{2}}{r^{5}}e^{{\rm i}\omega
r}(x-x^{\prime})(z-z^{\prime})dV^{\prime},$

where

$r=\sqrt{(x^{\prime}-x)^{2}+(y^{\prime}-y)^{2}+(z^{\prime}-z)^{2}}$

is the distance from the observation position ($\mathbf{x}$) to the source
point ($\mathbf{x}^{\prime}$). Then, we can obtain the average energy flux
according to Eq. (2)

$\displaystyle S_{x}$ $\displaystyle=$
$\displaystyle\frac{\rho_{a}g_{a\gamma}^{2}B_{e}^{2}}{16\pi^{2}\omega}\textrm{Re}\left[{\rm
i}I_{1x}(I^{\ast}_{2x^{2}}+I^{\ast}_{2y^{2}}+2I^{\ast}_{0})\right]$ (19)
$\displaystyle S_{y}$ $\displaystyle=$
$\displaystyle\frac{\rho_{a}g_{a\gamma}^{2}B_{e}^{2}}{16\pi^{2}\omega}\textrm{Re}\left[{\rm
i}I_{1y}(I^{\ast}_{2x^{2}}+I^{\ast}_{2y^{2}}+2I^{\ast}_{0})\right]$ (20)
$\displaystyle S_{z}$ $\displaystyle=$
$\displaystyle\frac{\rho_{a}g_{a\gamma}^{2}B_{e}^{2}}{16\pi^{2}\omega}\textrm{Re}\left[{\rm
i}(I^{\ast}_{2xz}I_{1x}+I^{\ast}_{2yz}I_{1y})\right].$ (21)

The component of the flux is defined as
$S_{x,y,z}=\langle\mathbf{S}\rangle\cdot\mathbf{e}_{x,y,z}$ . They can be
rewritten in the international unit, e.g., for $S_{x}$, that is

$\displaystyle S_{x}=1.76\times
10^{-20}\textrm{W}/\textrm{m}^{2}\frac{\rho_{a}}{0.3\textrm{GeV}/\textrm{cm}^{3}}\left(\frac{g_{a\gamma}}{10^{-12}\textrm{GeV}^{-1}}\right)^{2}$
$\displaystyle\frac{100\textrm{neV}}{m_{a}}\left(\frac{B_{e}}{10\textrm{T}}\right)^{2}\textrm{Re}\left[{\rm
i}\frac{I_{1x}}{\rm
m}(I^{\ast}_{2x^{2}}+I^{\ast}_{2y^{2}}+2I^{\ast}_{0})\right].$ (22)

Figure 6: The comparison between the energy flux (blue line) according to Eq.
(23) and its approximation in the long wave and far-field limit (red line) as
Eq. (19-21). The energy flux $S$ as a function of the observation distance for
$\omega=100\,$neV, $g_{a\gamma}=10^{-12}{\rm GeV^{-1}}$, $B=10\,{\rm T}$, and
$\rho_{a}=0.3\,{\rm GeVcm^{-3}}$. The approximation is consistent with the
numerical result when the condition $|\mathbf{x}|\gg|\mathbf{x}^{\prime}|$ is
satisfied.

When the observation is located at the far-field region
($|\mathbf{x}|\gg|\mathbf{x}^{\prime}|$) and the wave length is much larger
than the size of the radiation source ($\lambda\gg|\mathbf{x}^{\prime}|$), we
can easily get the magnitude of the flux from Eq. (5) as

$\displaystyle S$ $\displaystyle=$
$\displaystyle{\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}\omega^{2}V^{2}\over
16\pi^{2}r^{2}}{\rm sin}^{2}\theta$ (23) $\displaystyle=$ $\displaystyle
2.3\times
10^{-25}\textrm{W}/\textrm{m}^{2}\frac{\rho_{a}}{0.3\textrm{GeV}/\textrm{cm}^{3}}\left(\frac{g_{a\gamma}}{10^{-12}\textrm{GeV}^{-1}}\right)^{2}$
$\displaystyle\left(\frac{m_{a}}{100\textrm{neV}}\right)^{2}\left(\frac{B_{e}}{10\textrm{T}}\right)^{2}\left(\frac{V}{\rm
m^{3}}\right)^{2}\left(\frac{100\rm m}{r}\right)^{2}{\rm sin}^{2}\theta\,.$

Fig. 6 shows the comparison between the energy flux (blue line) according to
Eq. (19-21) and its approximation in the long wave and the far-field limit
(red line) as Eq. (23). For $\omega=100\,$neV, when $r\gtrsim 3\,$m, the
condition $|\mathbf{x}|\gg|\mathbf{x}^{\prime}|$ is satisfied, and the
approximation is consistent with the numerical result. For higher frequency or
axion mass, the integral function in Eq. (19-21) oscillates much faster, some
special numerical techniques are required and it will be left to subsequent
research.

##  III The experimental detection

Figure 7: The profile map of a detection scheme for radio-microwave radiated
photons ($\gamma$). The blue dashed lines represent the central axis. Four
detectors worked on different bandwidths surround the magnetic field to detect
the axion($a$)-induced radiation with four bandwiths: Band I (1-11 neV), Band
II (11-111 neV), Band III (0.111-1.11$\mu$eV), Band IV (1.11-11.11 $\mu$eV).
The receiver can be dish antennas. Each detector with a specific bandwidth can
consist of two identical self-detectors, as a small volume is conducive to
lowering the operating temperature.

We discuss the possible to detect the ADM-induced radiation for the homogenous
“cylindrical B-field”.

The radiated power increases with the volume or the side area of the magnetic
field, which however must be finite. For a fixed B-field volume or the side
area of the cylinder, the maximum of the power $P(\omega)$ occurs at the first
peak determined by $\omega\simeq\frac{3}{4R}\pi$
($R\omega=(n+\frac{3}{4})\pi>1$, n=0), before which the power $P(\omega)$ is
an increasing function of the frequency $\omega^{2}$, and after which
decreasing as $\frac{1}{\omega^{2}}$, e.g., see the bottom panel of Fig. 3. If
the maximum allowable radius $R$ is 2.5 m, the maximum value of the
$P(\omega)$ occurs at $\gtrsim$200 neV.

Therefore, we attempt to focus on the bandwidth of $1-10^{4}$ neV, which
corresponds to the frequencies $0.24\,{\rm MHz}-24\,{\rm GHz}$. At this
bandwidth, the typical detection level with techniques of the radio-
astronomical measurement is $\lesssim 10^{-22}$ W Horns:2013ira . For typical
parameters as in Eq. (11) and (12), the detectable axion-photon coupling
$g_{a\gamma}$ can reach about $10^{-14}\,{\rm GeV^{-1}}$ which is allowed by
the present observation except for a very narrow mass band excluded by ADMX
ADMX:2021nhd . It seems to be promising to detect the radiation from such
axions on experiment.

The thermal noise of the detecting instrument is the main noise. Within a
bandwidth of $\Delta\nu$, the ratio of the signal to one $\sigma$ fluctuation
in the noise, i.e, the signal to noise ratio (SNR), is given by Dicke’s
radiometer equation Dicke:1946glx :

${\rm SNR}={P_{\rm signal}\over T_{\rm n}}\sqrt{{\Delta t\over\Delta\nu}}$
(24)

where $T_{\rm n}$ represents the total noise temperature and $P_{\rm signal}$
is the ADM-induced radiated power.

###  III.1 broadband searches

Figure 8: The expected sensitivity (in shades of light blue) for our detection
schemes shown in Fig. 7. we assumes the signal-to-noise ratio SNR=5. The
height and radius of the cylindrical magnetic field are 1 m and 2 m,
respectively. The detection time $\Delta t=3\,$years for each detector, total
noise temperature $T_{\rm n}=1\,$K, $B=10\,{\rm T}$, and ADM energy density
$\rho_{a}=0.3\,{\rm GeVcm^{-3}}$ are also assumed. The dark shading shows
existing constraints. The lines of KSVZ Kim:1979if ; Shifman:1979if  and DFSZ
Dine1981 ; Zhitnitsky:1980tq  together orange strip are benchmark axion
predictions. Limit data for other experiments and plotting scripts are
available at Ref. Hare . The dashed black line marks the upper bound on ALP DM
Arias:2012az .

As we can see in Fig. 3 and 5, the radiation peaks present wide, especially at
low frequencies, since our scheme gives up the resonance enhancement on the
ADM-induced signal. The measurement bandwidth $\Delta\nu$ of our scheme mainly
depends on detector technology. We can use dish antennas as the detectors,
which are widely used in radio astronomy. For modern detectors of radio
astronomy, the bandwidths $\Delta\nu$ can be in excess of 1 GHz and the
spectral resolutions can be better than $10^{6}$ Horns:2013ira . We
conservatively divide the interested detection bandwidth $1-10^{4}\,$neV into
four section: Band I (1-11 neV), Band II (11-111 neV), Band III
(0.111-1.11$\mu$eV), Band IV (1.11-11.11 $\mu$eV). Since the radiometry device
of cylinder is very large and has symmetry with respect to the x-y plane and
z-axial, we can use four detectors with different bandwidths surrounding the
magnetic field to detect the radiation at the Band I-IV simultaneously . Each
detector occupies about a quarter of a sphere in space, see Fig. 7. The ideal
detectable power for each detector is a quarter of total power

$P_{\rm signal}=\frac{1}{4}P.$ (25)

Note that the detectors should closely surround the magnetic field to avoid
radiation energy leaking due to the diffraction effect. Each detector with a
specific bandwidth can consist of two identical self-detectors, as a small
volume is conducive to lowering the operating temperature.

###  III.2 The expected sensitivity

Using Eq. (8), (24) and (25), the expected sensitivity for our detection
scheme (Fig. 7) is shown in shades of light blue in Fig. 8. The detection time
is assumed to $\Delta t=3\,$years, SNR=5, total noise temperature $T_{\rm
n}=1\,$K, $B=10\,{\rm T}$, and $\rho_{a}=0.3\,{\rm GeVcm^{-3}}$. The height
and radius of the cylindrical magnetic field are 1 m and 2 m, respectively.

In general, the sensitivity curve shows a shape with low middle and high ends,
which is mainly determined by the radiated power. The strongest limitation can
reaches $g_{a\gamma}\simeq 10^{-14}\rm GeV^{-1}$ at about 500 neV, as the
maximum of the radiation power occurs at this mass point. The dark shading
shows existing constraints. Our sensitivity is far beyond existing constraints
except for ADMX which operates in the narrow bandwidth mode. If our scheme
also only focus on a very narrow bandwidth in which the frequency
corresponding to the maximum power is located by adjusting the radius $R$ of
the B-field to satisfy $\omega\sim\frac{3}{4R}\pi$, the sensitivity is
expected to increase by an order of magnitude.

##  IV Discussion

We mainly discuss the comparison of our scheme with other experiments,
especially the dish-antenna experiment that is similar to ours and the high
mass axion detections.

Figure 9: The profile map of detection scheme for infrared (about meV,
corresponds to 0.24 THz) radiated photons. The blue dashed lines represent the
central axis. Two parabolic mirrors surround the magnetic field and reflect
the incident ADM-induced radiation to the receivers at the focal point. This
scheme is similar to the BRASS experiment BRASShppt .

In the dish-antenna or cylindrical metal barrel experiment, the emitted power
from the dish surface is $P=\frac{1}{2}\rho_{a}g_{a\gamma}^{2}B_{\rm
e}^{2}\frac{A_{\rm dish}}{m_{a}^{2}}$ BREAD:2021tpx . It is similar to our
result $P=\frac{1}{2}\rho_{a}g_{a\gamma}^{2}B_{\rm e}^{2}\frac{A}{m_{a}^{2}}$
in Eq. (LABEL:pJ) using $\langle\rm{cos^{2}}\rangle=\frac{1}{2}$. When the
side area $A$ of the B-field parallel to $\mathbf{B}$ is equal to $A_{\rm
dish}$, the two powers are the same. It seems that the photon from axion
conversion is emitted from the interface between the magnetic field and the
outside. But, in our scheme, when $R\omega\ll 1$, the power in proportional to
the square of the volume and the mass so that the maximum power $P(\omega)$
for $5\gtrsim R\gtrsim 0.5\,$m appears at $0.1\lesssim\omega\lesssim
1\,\mu$eV. In this sense, our scheme is more suitable to detect the axions at
$0.1\lesssim\omega\lesssim 1\,\mu$eV band.

Compared to the LC circuit (e.g., Ouellet:2018beu ) and cavity experiments
(e.g., ADMX:2021nhd ), our detection scheme is simpler but requires a much
larger volume of B-field. It seems more reasonable to exploit the detection of
the axion-induced radiation by using some other particular experiments (e.g.,
stellarators) with a very strong and large B-field. The main challenge is to
detect the un-concentrated radiant energy in very low temperature mode.

Though the proposed detection mass in our scheme is in $1-10^{4}\,$neV band,
it is not constrained by resonance requirements as in cavity detections. At
higher frequency, the diffraction effect of radiation can be weakened so that
we can focus the radiation to a small detector according to geometric optics.
Therefore, it could, in principle, be used to detect axion with mass
$m_{a}>100\,\mu$eV. Fig. 9 shows the detection scheme for infrared radiated
photons. Two parabolic mirrors surround the magnetic field and reflect the
incident ADM-induced radiation to the receivers at the focal point. The energy
flux can be calculated by Eqs. (19)-(21).

##  V Conclusion

In summary, we have studied the direct radiation excited by the oscillating
axion DM in a homogenous magnetic field and tried to devise a scheme to detect
such axions.

We have concretely derived the analytical expression of the axion-induced
radiated power for cylindrical uniform magnetic field, see Eqs. (8)-(12). In
the long wave limit $R\omega\ll 1$, the radiation power is proportional to the
square of the volume and the axion mass $m_{a}$. When $R\omega\gtrsim 1$, the
target radiations oscillate and their peak powers are proportional to the side
area and inversely to $m_{a}^{2}$. The maximum power locates at
$\omega\sim\frac{3\pi}{4R}$ a for fixed radius $R$. Therefore, the axion in
the mass range $1-10^{4}$ neV is suitable for detecting for a finite volume of
the magnetic field. The radiated energy flux is also derived in Eqs.
(19)-(23).

Finally, we have discussed a scheme to detect the axions in this mass range
using four detectors of different bandwidths surrounding the B-field, see Fig.
7. Our expected sensitivity under the typical parameter values is far beyond
the existing constraints except for ADMX experiment, see Fig. 8.

More advanced and realistic detection schemes and techniques can be further
developed in the future, and our results are useful for detecting the ADM in
strong man-made or natural magnetic fields.

###### Acknowledgements.

We thank Seishi Enomoto, Chengfeng Cai, and Yi-Lei Tang for useful discussions
and comments. This work is supported by the National Natural Science
Foundation of China (NSFC) under Grant No. 11875327, the Fundamental Research
Funds for the Central Universities, China, and the Sun Yat-sen University
Science Foundation.

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brassweb.htm$\\#$note02.


# Unified Open-Domain Question Answering
with Structured and Unstructured Knowledge

Barlas Oğuz1, Xilun Chen1, Vladimir Karpukhin1,
Stan Peshterliev1, Dmytro Okhonko1, Michael Schlichtkrull2,3,
Sonal Gupta1, Yashar Mehdad1, Wen-tau Yih1
1Facebook AI Research, 2University of Amsterdam, 3University of Edinburgh,
{barlaso, xilun, vladk, stanvp, oxo, sonalgupta, mehdad<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Work done while interning with Facebook AI Research.

###### Abstract

We study open-domain question answering (ODQA) with structured, unstructured
and semi-structured knowledge sources, including text, tables, lists, and
knowledge bases. Our approach homogenizes all sources by reducing them to
text, and applies recent, powerful retriever-reader models which have so far
been limited to text sources only. We show that knowledge-base QA can be
greatly improved when reformulated in this way. Contrary to previous work, we
find that combining sources always helps, even for datasets which target a
single source by construction. As a result, our unified model produces state-
of-the-art results on 3 popular ODQA benchmarks.

##  1 Introduction

Figure 1: Unified multi-source question answering architecture: dense index
retrieves Wikipedia passages, tables and knowledge base relations.
Heterogeneous context is encoded independently through the encoder, then
processed jointly in the decoder, which generates the answer.

Answering any factual question has long been an inspirational challenge to
information retrieval and artificial intelligence researchers Voorhees (1999);
Lopez et al. (2011). Users can ask about any topic and the answer can be found
in any information source. Defined as such, the challenge of _open domain
question answering_ is extremely broad and complex. Though there have been
successful undertakings which embrace this complexity (notably Ferrucci,
2012), most recent works make simplifying assumptions as to the source of
answers. These fall largely in two categories: _structured data_ and
_unstructured text_.

A long line of research aims to answer user questions using a structured
_knowledge base_ (KB) (Berant et al., 2013; Yih et al., 2015). Typically, a KB
can be viewed as a knowledge graph consisting of entities, properties, and a
pre-defined set of relations between them. A question can be answered,
provided that it can be expressed within the language of relations and objects
present in the knowledge graph. With a high-quality, carefully curated KB,
answers can be extracted with fairly high precision. KBQA, however, struggles
with low answer coverage due to the cost to curate an extensive KB, as well as
the fact that many questions simply cannot be answered using a KB if the
answers are not entities.

The second line of work targets a large collection of unstructured text (such
as Wikipedia) (Chen et al., 2017) as the source of answers. While the
unstructured text is infinitely varied, open-domain question answering from
text (TextQA) has advanced greatly in recent years (Yang et al., 2019; Lee et
al., 2019; Karpukhin et al., 2020; Guu et al., 2020; Izacard and Grave, 2020),
thanks to recent advances in machine reading comprehension and text retrieval.

When answering questions using both structured and unstructured information, a
straightforward solution is combining specialized TextQA and KBQA systems. The
input question is sent to multiple sub-systems, and one of them is selected to
output the final answer. However, while this approach may take advantage of
the state-of-the-art models designed for different information sources, the
whole end-to-end system becomes fairly complex. It is also difficult to handle
questions that can only be answered when reasoning with information from
multiple sources is required.

Having a more integrated system design that covers heterogeneous information
sources has proven to be difficult. One main reason is that techniques used
for KBQA and TextQA are drastically different. The former exploits the graph
structure and/or semantic parsing to convert the question into a structured
query, while TextQA has mostly settled on the retriever-reader architecture
powered by pre-trained transformers. In an attempt to preserve structure,
recent work on multi-source QA has tried to incorporate free text into graph
nodes. While this approach makes it amenable to KBQA methods (Sun et al.,
2018; Lu et al., 2019), the observed performance gain remains limited.

In this work, we propose an alternative approach for open-domain question
answering with heterogeneous information sources. Instead of having multiple
specialized sub-systems or incorporating text into knowledge graphs, we
flatten the structured data and apply TextQA methods. Our main motivation for
doing so is to make the powerful machinary of pre-trained transformers
available for structured QA. Such models have recently been shown to excel at
combining information from a large number of text documents (Izacard and
Grave, 2020), and are thus a natural candidate for source fusion as well.
There are also other advantages of this approach — it opens the door to a
simple, unified, single-model architecture. We can easily support semi-
structured sources such as lists and tables, as well as fully structured
knowledge bases. Moreover, there is no need to specially handle the schema or
ontology that define the structure of databases.

We first verify our approach by modeling KBQA as a pure TextQA task. We
represent all relations in the KB with their textual surface form, and train a
_retriever-reader_ model on them as if they were text documents. This simple
approach works incredibly well, improving the exact match score on the WebQSP
dataset by $11\%$ over previous state of the art. This result further
justifies our choice of unifying multi-source QA under the TextQA framework as
it can improve KBQA performance _per se_.

We consider lists, tables, and knowledge bases as sources of structured
information. For our multi-source QA experiments, each of them are converted
to text using simple heuristics. We model various combinations of structured
sources with text, and evaluate on four popular open-domain QA datasets,
ranging from entity-heavy KBQA benchmarks to those targeting free form
sources. Our results indicate that while the best single source of information
varies for each dataset as expected, combining sources improves results in all
cases. In this way we obtain state-of-the-art results for 3 datasets,
advancing the published art on NaturalQuestions by 3.7 points and on
WebQuestions by 2.7 points.

In addition, we consider the multi-task setting in which the source of
questions is not known a priori, as would be the case for a practical system.
We train a _unified_ model on all datasets, and show that it can perform well
across this diverse set of questions. Our best system incorporates some 27
million passages composed of text and lists, 455,907 Wikipedia tables, and
billions of relations from the Wikidata knowledge base in a single, unified
open-domain QA model.

##  2 Background & Related Work

###  2.1 Knowledge-base question answering

A knowledge base is composed of subject-predicate-object triplets
$(e_{1},p,e_{2})$, where $e_{1},e_{2}\in\mathcal{E}$ are _entities_ or _nodes_
, and $p$ is a predicate which describes the binary relation between $e_{1}$
and $e_{2}$. Nodes can be named entities such as place or person names, or
unnamed entities such as dates or numbers. Examples of relations are
_capital_city_ , _married_to_ etc.

Modern KBs such as Freebase (Bollacker et al., 2008), DBPedia (Auer et al.,
2007), and Wikidata (Vrandečić and Krötzsch, 2014) can contain billions of
triplets, describing thousands of relations between millions of entities.
These are human curated, highly accurate collections of information, and have
served as a rich knowledge source for question answering systems.

The prevailing approach for answering questions from a KB is semantic parsing
(Berant et al., 2013; Yih et al., 2015), where a natural language question is
converted into a logical form that can be used to query the knowledge base.
Such methods explicitly utilize the graph structure of the KB, and as such are
not immediately transferable to other knowledge sources.

###  2.2 Question answering from tables

Large amounts of authoritative data such as national statistics are often only
available in the form of tables. Even for simple, natural questions asked by
users of a search engine, a significant fraction of them can be answered from
tables (Kwiatkowski et al., 2019). Surprisingly, while KBQA and TextQA has
enjoyed increasing popularity, tables as a source of information has escaped
the attention of the community save for a few recent works.

Working with web tables can be challenging, due to the lack of formal schema,
inconsistent formatting and ambiguous cell values (e.g. entity names). In
contrast to relational databases and KBs, tables can at best be described as
_semi-structured_ information. (Sun et al., 2016) considered open domain QA
from web tables, however makes no use of unstructured text. Some recent work
investigates MRC with tables without a retrieval component (Pasupat and Liang,
2015; Yin et al., 2020; Chen et al., 2019). In addition, (Chen et al., 2020a,
b) investigates open domain QA using tables and text. While they’re in a
similar direction, these works focus on complex, crowd-sourced questions
requiring more specialized methods, while we target the case of simple,
natural questions and investigate if popular TextQA and KBQA benchmarks can be
further improved with the addition of tables.

###  2.3 Open-domain question answering

KBQA is ultimately limited in its coverage of facts and the types of questions
it can answer. On the other hand, large collections of text such as Wikipedia
or CommonCrawl promise to be a richer source of knowledge for truly open
domain question answering systems. This line of work (which we will refer to
as TextQA) has been popularized by the TREC QA tracks (Voorhees, 1999), and
has seen explosive growth with the advent of neural machine reading (MRC)
(Rajpurkar et al., 2018) models. In the neural era, Chen et al. (2017) were
the first to combine MRC with retrieval for end-to-end QA. Subsequent work
cemented this _retriever-reader_ paradigm, with improved reader models (Yang
et al., 2019; Izacard and Grave, 2020) and neural retrievers (Lee et al.,
2019; Guu et al., 2020; Karpukhin et al., 2020). Despite impressive advances,
TextQA systems can still under-perform KBQA, especially on benchmarks
originally created for KBs. They also fall short of universal coverage,
because they ignore other (semi-)structured information sources such as
tables.

###  2.4 Fusion of text and knowledge-base

As discussed, KBQA and TextQA are intuitively complementary, and several
attempts have been made to merge them to get the benefits of both. An early
example is Ferrucci (2012), which combines multiple expert systems and re-
ranks them to produce the answer. More recent work attempts to enrich the KB
by extracting structure from text. One way to accomplish this is using OpenIE
triplets (Fader et al., 2014; Xu et al., 2016), thus staying completely within
the semantic parsing paradigm. Somewhat closer to our approach are
UniversalSchemas (Riedel et al., 2013; Das et al., 2017), which embeds KB
relations and textual relations in a common space. Yet, UniversalSchemas are
also constrained to an entity-relation structure. The latest in this line are
the works of Sun et al. (2018, 2019), which augments the knowledge graph with
text nodes and applies graph methods to identify candidate answers.

By retaining structure, previous work was able to take advantage of KBQA
methods, but also fail to capture the full richness of TextQA. We depart
radically in our approach, by foregoing all structure, and directly applying
TextQA methods based on the more general _retriever-reader_ architecture. We
also evaluate on a more diverse benchmark set composed of natural open domain
datasets, as well as those originally meant for KBQA, and demonstrate strong
improvements in this truly open-domain setting.

##  3 Modeling

###  3.1 Retriever-reader architecture

We use a retriever-reader architecture, with _dense passage retriever_ (DPR)
(Karpukhin et al., 2020) as retriever and _fusion-in-decoder_ (FiD) (Izacard
and Grave, 2020) as our reader, which is a seq-to-seq generative transformer.
The dense index includes lists, tables and KB relations, as well as regular
text passages. Retrieved documents are thus _heterogeneous_ , composed of a
mix of all sources. Each retrieved document is concatenated with the question,
then independently encoded by the reader encoder. Fusion of information
happens in the decoder, which computes full attention over the entire
concatenated input representations. The overall architecture is illustrated in
Figure 1.

##### Dense retrieval

DPR consists of a dense document encoder and a question encoder, trained such
that positive documents have embeddings which are close to the question
embedding in dot product space. We follow the original DPR implementation
closely, starting from BERT-base (Devlin et al., 2019) encoders, using
100-token text passages, a single negative document per question while
training and the same hyper-parameters. We further include tables, lists and
KB relations in the index. The details of how these are processed into
documents and merged are in the subsequent sections.

One improvement we make to the training process is iterative training, where
better hard negatives are mined at each step using the model at the previous
step, similar to (Xiong et al., 2020a). All our models, including our text-
only baselines benefit from this change. We find 2 iterations to be
sufficient.

##### Fusion in decoder

The FiD reader has been demonstrated to have strong performance in the text-
only setting and to be effective in fusing information from a large number of
documents. We thus find it to be a natural candidate for source-fusion. We use
FiD model with T5-large (Raffel et al., 2019), 100 retrieved documents as
input and the originally published hyper-parameters for all experiments.

###  3.2 KBQA with retriever-reader

In this section, we describe our approach to KBQA, which is to reduce it to
TextQA and to apply our retriever-reader model to tackle it. As we will show
in Section 4, this approach works surprisingly well.

Figure 2: Converting Freebase and Wikidata relations to text.

In order to apply our retriever-reader model, we first convert KB relations
into text using very simple heuristics. For a relation triple $\langle
subj,pred,obj\rangle$, where $subj$, $pred$ and $obj$ are the subject,
predicate and object of the relation respectively, we serialize it by
concatenating the text surface forms of $subj$, $pred$ and $obj$.

More complex relations involve multiple predicates and objects (sometimes
known as _hyper-relations_), such as _Natalie Portman played the character
Padmé Amidala in the movie Star Wars_ , and can be expressed differently
depending on the KB. In particular, Freebase uses _compound value types_
(CVTs) to convert a hyper-relation into multiple standard triples, while
Wikidata allows a predicate to have _qualifiers_ to express additional
properties Pellissier Tanon et al. (2016). In this work, we convert a hyper-
relation into a single sentence by forming a separate comma-separated clause
for each predicate (Figure 2). A side benefit of this approach is that these
complex relations are now represented as a single piece of text, whereas they
would normally be considered multi-hop and require more complex methods (Fu et
al., 2020) if using traditional graph-based KBQA models.

The next step is to retrieve relevant relations for a given question. One
approach is to index every (hyper-)relation of the KB in a dense index, which
can then be queried with the DPR-encoded question vector. Unfortunately, the
number of relations in a modern KB are in the _billions_ , which prohibits the
straight forward application of this method. Therefore, we employ an Entity
Linking (EL) system (STAGG (Yih et al., 2015) in the case of Freebase and ELQ
(Li et al., 2020) for Wikidata) to reduce the number of relations we need to
embed and index for each question. Specifically, we obtain the 2-hop KB
neighborhood of the linked entities (the portion that can be reached by
traversing at most 2 edges from a linked entity) in each question, and use DPR
to retrieve the top K relations from this set. Since each relation is a single
sentence, we combine consecutive relations (ranked by DPR similarity) into
longer documents of at most 100 tokens (similar to the TextQA documents)
before we feed them to the FiD reader. Finally, FiD is run in the same fashion
as in TextQA where the _context documents_ for each question now consist of
relation texts generated as mentioned above.

###  3.3 Tables

English Wikipedia contains more than 3 million tables (‘classical’ tables
embedded in text as well as specialized tables like info-boxes) which are a
huge source of factual knowledge by themselves. The answer to approximately a
quarter of the questions (that have short answers) in the NaturalQuestions
(NQ) dataset can be found in Wikipedia tables. These tables, however, have
largely been ignored in recent open-domain QA work since it usually requires a
separate dedicated model to reason over table structure. On the other hand,
many table-related questions in NQ do not require complex reasoning — it often
suffices to understand the table’s column values and a single row in order to
answer the question. We hence propose a simple approach to serialize tables
and incorporate them into our retriever-reader framework.

We collect all the passage candidates provided by the NQ training set which
are encoded as HTML tables. Wikipedia allows for nested tables, which is often
used in info-boxes. We extract nested tables from their ’parent’ tables and
treat them as separate items. We filter out ‘service’ tables that don’t carry
any information related to the page content, as well as single row tables,
resulting in a total of 455907 tables. In addition, we use about 19k training
samples from the NQ training set with short answer spans located in tables for
both retriever and reader training.

As with KB relations, semi-structured content in tables need to be
‘linearized’ into text for the _retriever-reader_ model to work. There are
many ways to do such linearization (see Yin et al., 2020; Chen et al., 2019).
We tried two types of tables linearization: ’template’-like encoding (similar
to the ’template’ scheme in Chen et al. (2019)) and a trivial one where cell
values are concatenated to form the rows’ text representation separated by the
newline token. We find the latter works the best (see Table 3) for dense
retrieval of tables. The simple linearization also works well with our reader,
which achieves a reading-comprehension score of 67.5 exact match on the gold
context tables subset of the NQ development set.

In order to use the linearized table representations in bi-encoder based
retriever training, one also needs to fit them into sequences of limited
length. A typical way of limiting the size of a table representation is to
sample rows or columns. We use row sampling to generate table samples as
follows:

  1. 1.

Take the first non-empty table row as the header and include it in every table
chunk.

  2. 2.

For generating positive training sapmles, take all rows containing any of the
answer spans while the total linearized sequence length is below the encoder
limit.

  3. 3.

If the sequence length is still below the limit, the rest of the rows are
randomly appended.

The heuristic to select the first non-empty row as header is crucial and adds
4-6 points to top-20 passage accuracy.

We mine hard negative samples for tables in the same way used in DPR — we
build an inverted index over all tables using an entire table as index
document and query Lucene with its BM25 implementation. We use the same row
sampling method described above for these hard negative samples but without
step #2. Thus we obtain an extra 19k training samples with linearized tables
as positive and hard negative passages and mix this data with the ’regular’ NQ
training set during training.

###  3.4 Combination of sources

We compare 5 variations of our model, each with a different combination of
information sources. We have _Text_ -only, _Tables_ -only and _KB_ -only
variants as single-source baselines. Next, the _Text* + tables_ model makes
use of the entire Wikipedia dump, including lists and tables. Finally we add a
KB resulting in the _Text* + tables + KB_ model.

The _Text* + tables_ model uses a unified dense index where text passages and
table chunks are jointly indexed. For the _Text* + tables + KB_ model, since
KB relations cannot be naturally chunked into 100-token documents for
retrieval, we index them separately and then merge results with a fixed
allocated quota for KB relations. This quota is determined to maximize
retrieval recall on the development set.

###  3.5 A unified model

Multi-tasking can be useful to improve the generalizability of models, and to
exploit beneficial transfer to tasks with comparatively smaller training data.
For question answering, previous work has investigated multi-task models in
the context of retrieval (Karpukhin et al., 2020; Guo et al., 2020), and
reading comprehension (Talmor and Berant, 2019; Fisch et al., 2019).

To our knowledge, no such study has been done for an end-to-end open domain QA
model. For our purposes, such a study is especially important, since in a
realistic setting, the best information source to answer a users question is
unknown _a priori_. Therefore we also experiment with a setting where we train
a single model on the combination of all 4 datasets and evaluate without any
input to the model as to the source of questions.111We lowercase questions and
remove question marks to ensure this is the case as much as possible. We refer
to this as the _unified_ setting. We train unified models for all 5 variants
described above. The smaller datasets, WebQ and TREC, are upsampled 5 and 8
times respectively while training.

##  4 KBQA as TextQA: A Motivating Experiment

In this section, we present a motivating experiment showing that our unified
QA approach not only provides a natural pathway to multi-source ODQA, but also
improves KBQA per se. In particular, we evaluate our approach on a widely-used
KBQA dataset, WebQSP Yih et al. (2016), in the single-source setting.

We use Freebase as the knowledge source, and re-use pre-computed STAGG entity
linking results and 2-hop neighborhoods as provided by Sun et al. (2018). We
then convert KB relations in the 2-hop neighborhood into text, retrieve the
most relevant ones using DPR to form 100 context passages, and feed them into
the T5 FiD reader as described in Section 3.2. The results are shown in Table
1, where the numbers represent _Hits@1_ , or the percentage of the model’s
top-predicted answer being a “hit” (exact match) against one of the gold-
standard answers.

We see that our KBQA method outperforms previous state-of-the-art methods by a
wide margin, resulting in a $34\%$ error rate reduction and improving exact
match accuracy to $79.1\%$. Since we adopt the exact same KB setup and pre-
processing from previous work, this improvement can be attributed purely to
the _retriever-reader_ model. We take this result as strong evidence for our
claim that powerful TextQA methods generalize well to structured data, and
offer a natural new framework for unifying structured and unstructured
information sources.

Model | Hits@1
---|---
GraftNet Sun et al. (2018) | 67.8
PullNet Sun et al. (2019) | 68.1
Our KBQA (T5-base) | 76.7
Our KBQA (T5-large) | 79.1
Table 1:  Hits@1 on WebQSP dataset using Freebase.

##  5 Multi-Source QA Experiments

Model | NQ | WebQ | Trivia | TREC | Avg.
---|---|---|---|---|---
SoTA | 51.41 | 55.13 | 67.61 | 55.32 | 57.3
Per-dataset models |  |
Text | 49.0 | 50.6 | 64.0 | 54.3 | 54.5
Tables | 36.0 | 41.0 | 34.5 | 32.7 | 36.1
KB | 27.9 | 55.6 | 35.4 | 32.4 | 37.8
Text + tables | 54.1 | 50.2 | 65.1 | 53.9 | 55.8
Text + tables + KB | 54.0 | 57.8 | 64.1 | 55.3 | 57.8
Unified model |  |
Text | 50.3 | 45.0 | 62.6 | 45.7 | 50.9
Tables | 34.2 | 38.4 | 33.7 | 31.1 | 34.4
KB | 25.9 | 43.3 | 34.2 | 38.0 | 35.4
Text + tables | 54.6 | 44.3 | 64.0 | 48.7 | 52.9
Text + tables + KB | 53.7 | 56.9 | 63.4 | 51.3 | 56.3
Table 2: Exact match results on the test set. SoTA numbers are from (Izacard
and Grave, 2020)1, (Iyer et al., 2020)2 which are TextQA approaches, and Jain
(2016)3, which is a KBQA method. Jain (2016) reports another metric, however
their predictions are available from which we calculated the EM score.

With the encouraging results on KBQA, we now go beyond a single information
source and present our main experiments on unified multi-source question
answering.

###  5.1 Datasets

For our main experiments, we use the same datasets that have recently become
somewhat standard for open-domain QA evaluation (Lee et al., 2019), with the
exception of SQuAD (Rajpurkar et al., 2018):

NaturalQuestions (NQ) Kwiatkowski et al. (2019) consists of questions mined
from real Google search queries and answer spans annotated from Wikipedia
articles. While targeting mainly free form text, annotations also include
tables.

WebQuestions (WebQ) Berant et al. (2013) targets Freebase as source of
answers, and questions come from Google Suggest API.

TriviaQA (Trivia) Joshi et al. (2017) contains a set of trivia questions with
answers that were originally scraped from the Web.

CuratedTREC (TREC) Baudiš and Šedivỳ (2015) is a collection of questions from
TREC QA tracks and various Web sources, intended to benchmark open-domain QA
on unstructured text.

###  5.2 Results

Main results are presented in Table 2. In the first set of experiments, we
train a reader model independently for each dataset, as typically done in
previous work. We use Freebase as knowledge base for WebQuestions as intended,
and use Wikidata for all others.

Unsurprisingly, the best performing single-source model is the one each
dataset was designed for: text for NQ, Trivia, TREC and KB for WebQ. While the
best source combination varies from one dataset to another, fusing text with
at least one structured source improves results in _all_ cases, obtaining
state-of-the-art results for NQ, WebQ and TREC. Tables-only model also has
non-trivial performance and performs similarly for all datasets.

In the _unified_ setting, where a single model is trained across all datasets,
we observe similar trends with respect to sources, however performance is
generally lower, especially for small datasets (WebQ and TREC). We note that
T5 is already pre-trained with question answering tasks, which might be why no
gain is observed from multi-tasking. More importantly, we observe clear
improvements from combining sources, with the _Text* + tables + KB_ model
outperforming the _Text_ -only baseline by 5.4 points on average in this
realistic setting.

###  5.3 Analysis

Having established the value in combining sources, we now turn to the question
of _how_ adding additional sources improves results.

####  5.3.1 Retrieval

The natural assumption is that adding more data improves the _recall_ of
answers, thereby improving end-to-end performance. We verify this assumption
in the top half of Table 3, where we present retrieval results on the NQ
development set. We clearly see that adding additional sources improves
retrieval accuracy, which can be correlated with end-to-end accuracy as
presented in table 2.

Model | R@20 | R@100
---|---|---
Text-only | 80.0 | 85.9
w\lists | 82.7 | 89.6
w\tables | 83.1 | 91.0
w\lists + tables | 85.0 | 92.2
w\lists + tables + KB | 83.4 | 92.8
Tables-only |
simple linearization | 86.3 | 94.3
template linearization | 60.8 | 69.4
Table 3: Retrieval recall on the NQ dev set with different settings. All non
text only results are given for the model trained with NQ table samples.
Tables only results are for a NQ dev subset which has answers in tables.

####  5.3.2 Reader

We investigate if the observed improvements are indeed due to the reader
extracting answers from contexts retrieved from the additional information
sources. We ompare a baseline model with a candidate model with additional
sources, $\mathcal{S}$. Unfortunately, due to the nature of FiD generative
reader, it is not trivial to ascertain which input document(s) contributed the
answer. As a proxy, we look at the source type of _positive_ context documents
(those which contain a correct answer string) feeding into the candidate
model. Let $\mathcal{Q}$ denote the set of questions in the development set,
and $\mathcal{Q^{\prime}}$, the improvement set where the candidate model
outperforms the baseline model. If the improvement is due to additional
sources, we would expect fractionally more questions to have positive
documents of type $\mathcal{S}$ in $\mathcal{Q^{\prime}}$ as compared with
$\mathcal{Q}$. In Figure 3, we show this analysis for two cases. The first one
compares a baseline _text_ -only model to a model with lists and tables added
on NQ. The second one compares a _text*+tables_ model with _text*+tables+KB_
on the WebQ dataset. In both cases, we indeed see that the additional
information (lists and tables in the first case, and KB relations in the
second case) contribute disproportionally in the improvement set.

While this analysis shows that increased recall from additional sources does
contribute to the improvement, it does not immediately follow that the
improvement is due to this alone. There are other factors which might also
contribute, such as the addition of varied sources improving model quality
overall, or additional training samples from the tables set contributing as
well. To see how much of the gain might be due to these indirect factors, we
evaluate the _text*+tables_ model with text-only input. We find that this
achieves a similar performance (48.7 EM) on the NQ test set compared to a
_text_ -only model on the same input, suggesting that these other factors are
not a major contributor and that the improved performance is primarily due to
the added knowledge from structured sources.

Figure 3: Percentage of questions with answers in additional sources. For NQ
the additional sources are list and tables. For WebQ the additional source is
KB.

##  6 Discussion

We demonstrated a powerful new approach for unifying structured and
unstructured information sources for open-domain question answering. We adopt
the extremely simple and general _retriever-reader_ framework and showed not
only that it works for structured sources, but improves over traditional KBQA
approaches by a wide margin. By combining sources in this way, we achieved new
state-of-the art results for 3 popular open-domain QA benchmarks.

However, our model also has several shortcomings in its current form. As a
result of flattening all sources into text, we lose some desirable features of
structured knowledge bases: the ability to return all answers corresponding to
a query, and the ability to infer multi-hop paths to answer more complex
questions. In this work we have side-stepped the first issue by focusing on
the exact match metric (equivalent to Hits@1), which is standard in the open-
domain QA literature, but largely ignores multiple answers. We were also able
to ignore the second issue, since the datasets we evaluated on, while
standard, are composed mostly of simple, natural user questions which can be
answered from a single piece of information.

We do believe these are important details and they can be addressed within the
framework described here. For instance, outgoing edges of an entity with the
same relation can easily be merged, thus encoding all answer entities into a
single text representation. It’s also possible to simply generate multiple
answer candidates from the readers decoder. For multi-hop question answering,
there is recent work (Xiong et al., 2020b) successfully extending dense
retrieval to the multi-hop setting, which could naturally be applied within
our framework. It remains to be seen how these approaches would compare to
more traditional structured methods, and we leave this for future work.

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