SM

    Standard Model
CPV

    CP Violation
EH

    Euler-Heisenberg
BSM

    Beyond the Standard Model
COM

    Center of Mass
QED

    Quantum Electrodynamics
EFT

    Effective Field Theory
ALP

    Axion-Like Particle
LV

    Lorentz Violation
DM

    Dark Matter
EOM

    Equation of Motion
EP

    Equivalence Principle
AP

    Atomic Physics
DDM

    Direct Dark Matter
DP

    Dark Photon
EM

    Electro-magnetic
SSB

    Spontaneous Symmetry Breaking
EWSB

    electroweak symmetry breaking
EW

    electro-weak
VEV

    vacuum expectation value
FC

    fundamental constant
ULDM

    ultralight dark matter
QFT

    quantum field theory
NGB

    Nambu Goldstone Boson
GB

    Goldstone Boson
pNGB

    pseudo-Nambu Goldstone Boson
CW

    Coleman-Weinberg
RG

    Renormalization Group
pNGB

    pseudo Nambu-Goldstone boson
CP

    charge-parity
aainstitutetext: Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, 234 Herzl Street, Rehovot 7610001,
Israelbbinstitutetext: Department of Physics and Astronomy, University of
Delaware, Newark, Delaware 19716, USAccinstitutetext: Joint Quantum Institute,
National Institute of Standards and Technology and the University of Maryland,
Gaithersburg, Maryland 20742, USA

# The Phenomenology of Quadratically Coupled Ultra Light Dark Matter

Abhishek Banerjee ID a Gilad Perez ID b,c Marianna Safronova a Inbar Savoray
ID a Aviv Shalit ID

###### Abstract

We discuss models of ultralight scalar Dark Matter (DM) with linear and
quadratic couplings to the Standard Model (SM). In addition to studying the
phenomenology of linear and quadratic interactions separately, we examine
their interplay. We review the different experiments that can probe such
interactions and present the current and expected future bounds on the
parameter space. In particular, we discuss the scalar field solution presented
in [A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik and P. Wolf, Phys.Rev.D
98 (2018) 6, 064051], and extend it to theories that capture both the linear
and the quadratic couplings of the DM field to the SM. Furthermore, we discuss
the theoretical aspects and the corresponding challenges for natural models in
which the quadratic interactions are of phenomenological importance.

##  1 Introduction

One possible solution to the “missing mass” problem is that of an ultralight
sub-eV bosonic Dark Matter (DM) field, coherently oscillating to account for
the observed DM density (e.g. Catena:2009mf ; Graham:2013gfa ;
Arvanitaki:2014faa ; Graham:2015ifn ; Banerjee:2018xmn ). Such a light field
would oscillate with a frequency proportional to its mass $m_{\phi}$, and an
amplitude which is determined by $m_{\phi}$ and the DM density
$\rho_{\text{DM}}$,

$\displaystyle\langle\phi\left(t,\vec{x}\right)\rangle$
$\displaystyle\simeq\frac{\sqrt{2\rho_{\text{DM}}}}{m_{\phi}}\cos(m_{\phi}\,t+\delta)\,,$
(1.1)

where $\delta=m_{\phi}\,\vec{\beta}\cdot\vec{x}$ is some random phase with
$|\vec{\beta}|\sim 10^{-3}$ being the virial DM velocity. Possibly the most
simple model of an ultralight dark matter (ULDM) field is obtained by
augmenting the SM (of fundamental interactions and elementary particles) with
only one degree of freedom. Adding a free light spin-0 field, with its
misalignment angle appropriately tuned towards the end of inflation so that
its oscillation amplitude yields the right DM abundance, fully address the
missing mass problem (see for instance Kolb:1990vq  for more detail). Such a
model can be tested solely by its gravitational interactions Bar:2018acw ;
Bar:2019bqz ; Arvanitaki:2014wva , however, adding self-interactions may
render the DM distribution, and hence the corresponding bounds, non-robust.

More conceptually, models with spin-0 ULDM face two main theoretical
challenges. The first is associated with the hierarchy problem, namely, the
challenge of keeping the scalar light, although in the presence of
interactions microscopic quantum fluctuations are generically expected to
contribute dramatically to its mass. We shall discuss it further in the
following sections. The second challenge is associated with the fact that all
fields interact via gravity. Even in the absence of a direct coupling of the
scalar to the SM fields, one may argue that if the spin-0 particle is an
elementary, point-like, microscopic field, “gravity-mediated” interactions
will inevitably generate such an effective coupling. Below the Planck scale,
an Effective Field Theory (EFT) describing the SM elementary fields and the
new spin-0 elementary field would consist of (local) interaction terms between
the ULDM and the SM, suppressed by the Planck scale. Such a coupling may be
eliminated if the field is composite or in the presence of additional discrete
symmetries (see Dine:2022mjw ; Banerjee:2022wzk  for recent
discussions).111This can be thought of as a generalized version of the axion
quality problem Kamionkowski:1992mf ; Barr:1992qq ; Davidi:2017gir ;
Davidi:2018sii , (for a more general discussion see Calmet:2021iid ;
Perez:2020dbw , and Choi:1985cb ; Kim:1984pt ; Contino:2021ayn ; Perez:2020dbw
; Banerjee:2022wzk  for models that address the quality problem). In some more
detail, the QCD axion quality problem is attributed to the fact that the axion
potential resulting of QCD instanton-corrections can be disrupted by the
presence of Planck suppressed operators that do not respect the Pecci-Quinn
Symmetry, see for instance Dine:2012sla ; Hook:2018dlk  for reviews on the
topic.

Let us first consider the case where the DM field is elementary and there are
no additional discrete symmetries. It is interesting to note that for ULDM
models with scalar-linear couplings between the DM and the SM fields, any
masses below roughly 10-6 eV are excluded by experiments testing the
Equivalence Principle (EP) (corresponding to regions with effective coupling
$d^{(1)}_{x}\gtrsim 1$, see text around Fig. 9 for more details). Along this
line, we point out that one can identify a set of models where the EP bound is
greatly ameliorated or absent (one example for such a model is a pure dilaton,
but there are others, see Oswald:2021vtc ; WRESL2 ). In this case, a weaker
bound associated with fifth-force searches can be enforced, excluding models
with scalar ULDM Planck suppressed couplings for ULDM masses below 10-10 eV.

In addition, there is a broad class of well-motivated models where the ULDM is
predicted to interact with the SM fields beyond merely Planck suppressed
couplings. Two prime examples are models where the ULDM is spin-0 but a
pseudo-scalar, Axion-Like Particle (ALP), or where it is a CP-even scalar.
ALPs naturally arise in theories where a global U(1) symmetry is spontaneously
broken, for instance, in Froggatt-Nielsen type of models Froggatt:1978nt  that
address the mass hierarchies (usually denoted as the flavor puzzle), models
which account for lepton number conservation Gelmini:1980re , models of QCD
axion solution to the strong CP problem Kim:1979if ; Shifman:1979if ;
Zhitnitsky:1980tq ; Dine:1981rt , models which solve the hierarchy problem
Graham:2015cka  or combinations of the above Ema:2016ops ; Davidi:2018sii ;
Davidi:2017gir ; Banerjee:2022wzk . As we have already mentioned, scalar ULDM
models are more involved as they are susceptible to naturalness problems,
however, two main options were described in the literature. In the first, the
ULDM mass is protected by either an approximate scale-invariance symmetry
Arvanitaki:2014faa  or a discrete $\mathbb{Z}_{N}$ symmetry Brzeminski:2020uhm
. In the second, inspired by the relaxion paradigm Graham:2015cka , the ULDM
is an exotic type of ALP Banerjee:2018xmn , with its associated shift symmetry
and charge-parity (CP) invariance broken by two independent sectors
Flacke:2016szy ; Choi:2016luu ; Banerjee:2020kww . In all of the above models
but the last one, the ULDM couplings to the SM fields are, to leading order,
dominated by the ULDM derivative couplings to the appropriate SM current
operators, with the strength of these interactions dictated by the
transformations of the SM fields under the ULDM shift-symmetry. Generically,
we expected the dominant interaction to be linear with the DM field.

Most of the theoretical and experimental effort has been put towards studying
the linear-DM-SM interactions (see Hees:2018fpg ; Lin:2019uvt ;
Alexander:2016aln  for a recent review and Refs. therein). However, as
mentioned above, both fundamental scalar and axion models suffer from a rather
severe quality problem, namely they do not provide sufficient protection
against operators that link the ULDM field with the SM ones (usually denoted
as irrelevant operators), even if they are suppressed by super-Planckian
cutoffs. It motivates us to consider cases where the quadratic interactions
dominate over the linear ones, at least when considering the more severely
constrained scalar SM-operators. One example of such a scenario is in the case
of ultralight QCD axions, where the product of the mass, $m_{\phi}$, and the
decay constant, $f$, satisfies $m_{\phi}f\ll\Lambda_{\rm QCD}^{2}$, where
$\Lambda_{\rm QCD}$ is the dynamical scale of QCD. This can arise due to fine-
tuning, or in a particular type of $\mathbb{Z}_{N}$ models Hook:2018jle ;
DiLuzio:2021pxd . In such a case, quantum sensors looking for scalar-quadratic
oscillations of constants of nature, would be more sensitive to the presence
of ULDM QCD axion Kim:2022ype , compared with the well known conventional
searches which are based on magnetometer-type of quantum sensors. From the
properties of quantum field theory (QFT), and its low energy EFT perspective,
theories where the quadratic interactions of an ULDM scalar are experimentally
significant, seem to be exotic. First, it implies that the linear coupling
that should dominate the phenomenology is, for some reason, is suppressed.
Furthermore, the quadratic coupling of the ULDM, which is expected to generate
large additive contributions to the ULDM mass, is thus being pressured by
naturalness arguments. We nevertheless find it interesting to consider in
depth models where the phenomenology is dominated by the ULDM quadratic
coupling, in view of the interplay between the direct and indirect
experimental searches. Together with presenting the current and expected
future bounds on these models, we outline the experimental and theoretical
challenges associated with them, and study concrete examples in which these
challenges can be ameliorated.

The paper is organized as follows: in Subsections 1.1 and 1.2, we introduce
the ULDM models of interest and discuss their phenomenology, in particular,
the profile of quadratically coupled DM. In Section 2, we review the bounds
from different ULDM searches, considering current and future probes. In
Section 3, we study in detail the behavior of a quadratically coupled DM field
in the presence of a massive source such as the Earth. In addition, we comment
on the challenges of EP tests and Direct Dark Matter (DDM) searches due to the
DM field profile, and show how those can be addressed by performing
experiments in space. In Sections 4 and 5, we review the theoretical aspects
of models with sizable quadratic DM interactions with the SM, and provide
various examples in which these couplings are technically-natural, and
dominate over the linear ones. In Section 6, we study two specific models
solving the naturalness problem of the ULDM and allowing for a hierarchy
between the linear and quadratic coupling of the DM field. We conclude our
results in Section 7.

###  1.1 Model with linear DM couplings

We start by reviewing the case where the DM couples linearly to the SM fields.
Since we choose to focus on CP invariant theories, we distinguish between a CP
odd pseudo-scalar, $\phi(x)=a(x)$, and a CP-even scalar, $\phi(x)=\varphi(x)$.
The linear interactions can be characterized by the following low-energy
effective Lagrangians

$\displaystyle\mathcal{L}^{\varphi}_{\text{lin
scalar}}=\frac{d^{(1)}_{e}}{4M_{\rm
Pl}}\varphi\,F^{\mu\nu}F_{\mu\nu}-\frac{d^{(1)}_{g}\beta_{g}}{2M_{\rm
Pl}\,g}\varphi\,G^{b\mu\nu}G^{b}_{\mu\nu}-\frac{d^{(1)}_{m_{i}}}{M_{\text{Pl}}}\varphi\,m_{f}\psi_{f}\psi^{c}_{f}+\text{h.c.}\,,$
(1.2) $\displaystyle\mathcal{L}^{a}_{\text{lin pseudo-
scalar}}=\frac{\tilde{d}^{(1)}_{e}}{M_{\text{Pl}}}a\,F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{\tilde{d}^{(1)}_{g}}{M_{\text{Pl}}}a\,G^{b}_{\mu\nu}\tilde{G}^{b\mu\nu}-\frac{i\tilde{d}^{(1)}_{m_{i}}}{M_{\text{Pl}}}a\,m_{f}\psi_{f}\psi^{c}_{f}+\rm{h.c.}\,,$
(1.3)

where, $F_{\mu\nu}$ is the Electro-magnetic (EM) field strength,
$G^{b}_{\mu\nu}$ is the gluon field strength with color index $b$.
$\beta_{g}=-\left(\frac{11}{3}-\frac{2}{3}N_{f}\right)\frac{g^{3}}{16\pi^{2}}$
is the QCD beta functions, with $N_{f}$ being the number of light quarks.
$\psi_{i}$ ($\psi_{i}^{c}$) are the SM Weyl fermions (anti-fermion) with mass
$m_{f}$ ($f=u,d,e$ being a flavor index), $M_{\text{Pl}}=2.4\times
10^{18}\,{\rm GeV}$ is the reduced plank mass,

$\tilde{X}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}X_{\rho\sigma}$ with
$X=F,G^{b}$. $d^{(1)}_{i}$ and $\tilde{d}^{(1)}_{i}$ are dimensionless scalar
and pseudo-scalar linear DM couplings, respectively.222Eq. (1.3) is basis
dependent. One can perform pseudo-scalar dependent redefinition of the fermion
fields to remove pseudo-scalar coupling from the mass term or from a
topological term.
The analysis and the bounds on the scalar coupling, $d^{(1)}_{i}$, can be
found in Arvanitaki:2014faa ; Safronova:2017xyt . The oscillating background
of $\phi$, as given in Eq. (1.1), induces a temporal variation of the mass of
the SM fermions, the fine-structure constant $(\alpha)$, and the strong
coupling constant $(\alpha_{s})$ as

$\displaystyle Y_{i}(t)=Y_{i}(1+d_{i}^{(1)}\left<\phi\right>/M_{\rm Pl})$
(1.4)

where $Y\in(\alpha,\alpha_{s},m_{f})$. DDM searches are sensitive to the time
variation of fundamental constants, with their sensitivity at a given
frequency $\omega$ given as

$\eta^{\rm
DDM}\left(\omega\right)=\mathcal{F}_{\omega}\left(\left(\frac{\delta
Y_{i}}{Y_{i}}(t)\right)^{\rm DDM-lin}\right)\simeq\;\Delta\kappa_{i}\
\frac{d^{(1)}_{i}}{M_{\text{Pl}}}\mathcal{F}_{\omega}\left(\left\langle\phi\right\rangle\right)\;\simeq\;\Delta\kappa_{i}\
\frac{d^{(1)}_{i}}{M_{\text{Pl}}}\frac{\sqrt{2\rho_{\text{DM}}}}{m_{\phi}}\,.$
(1.5)

In equation (1.5), $\Delta\kappa_{i}\equiv\kappa^{A}_{i}-\kappa^{B}_{i}$ is
the difference of the sensitivity coefficients of a specific transition (see
e.g. Antypas:2019yvv ; Safronova:2017xyt  and refs. therein) and
$\mathcal{F}_{\omega}\left(f\left(t\right)\right)$ is the root of power
spectral density at frequency $\omega$, given by
$\mathcal{F}_{\omega}\left(f\left(t\right)\right)=\sqrt{\frac{1}{2\pi
T}\Big{|}\int^{T}_{0}f\left(t\right)e^{-i\omega t}dt\Big{|}^{2}}$ where $T$ is
total duration of the experiment.

In addition, the scalar field $\phi$ can also mediate long-range forces
between two masses. Therefore, another constraint on the parameter space of a
light scalar DM arises from experiments testing the EP or deviations from
Newtonian gravity Fischbach:1996eq . A linear $\phi$ coupling with the SM
generates a Yukawa force at tree-level, as shown in Figure 1.

The Yukawa poten tial that affects a test body $A$ in the presence of a
massive central body $C$, such as the Earth, has the following form

$V_{\text{Yukawa}}^{\text{lin}}\simeq-
Q_{i}^{A}Q_{i}^{C}\left(d_{i}^{(1)}\right)^{2}\dfrac{1}{4\pi
r}e^{-m_{\phi}r}\,,$ (1.6)

where $Q^{A}$ and $Q^{C}$ are the dilatonic charges of the test body and the
central body respectively, (see e.g Damour:2010rp ) and $r$ is the distance
between $A$ and $C$. The sensitivity of EP violation tests is characterized in
terms of the differential acceleration $a$ between two test bodies $A$ and $B$
in the presence of a source $C$, and takes the following form

$\eta^{\rm EP}=\left(\frac{\delta a_{\text{test}}}{a}\right)^{\text{EP-
linear}}\propto\;\left(d^{(1)}_{i}\right)^{2}\,\Delta Q_{i}\,Q_{i}^{C}\,,$
(1.7)

where $\Delta Q_{i}=Q_{i}^{\rm A}-Q_{i}^{\rm B}$.

Figure 1: A Yukawa potential is generated at tree level from the exchange of
$\phi$.

###  1.2 Model with quadratic DM couplings

The same analysis for the DDM and EP bounds on a linear $\phi$ theory can be
extended to quadratic interactions. We focus on a CP invariant quadratic
$\phi^{2}$ theory, given by the following effective Lagrangian

$\mathcal{L}_{\text{quad-
int}}\\!=\frac{\phi^{2}}{2}\left[\frac{d^{(2)}_{e}}{4M_{\text{Pl}}^{2}}F^{\mu\nu}F_{\mu\nu}-\frac{d^{(2)}_{g}\beta_{g}}{2M_{\rm
Pl}^{2}\,\,g}G^{b\mu\nu}G^{b}_{\mu\nu}-\frac{d^{(2)}_{m_{i}}}{M_{\rm
Pl}^{2}\,}m_{i}\psi_{i}\psi^{c}_{i}+\text{h.c.}\right],$ (1.8)

where the $d^{(2)}_{i}$ are the dimensionless quadratic couplings.

Since $\phi$ acquires a time-dependent vacuum expectation value (VEV) as in
Eq. (1.1), oscillations of fundamental constants will be generated similarly
to the linear theory, but with a background value of
$\left\langle\phi^{2}\right\rangle$ instead of
$\left\langle\phi\right\rangle$. For example, the variation of a constant
$Y_{i}$ can be written as,

$\frac{\delta
Y_{i}(t)}{Y_{i}}\simeq\;\Delta\kappa_{i}\frac{d^{(2)}_{i}}{2M_{\text{Pl}}^{2}}\left\langle\phi^{2}\right\rangle\simeq\Delta\kappa_{i}\frac{d^{(2)}_{i}}{M_{\text{Pl}}^{2}}\frac{\rho_{\text{DM}}}{m_{\phi}^{2}}\cos^{2}\left(m_{\phi}t+\delta\right)$
(1.9)

One should note that since the temporal modulation of the fundamental constant
should follow a $\cos^{2}\left(m_{\phi}t\right)$ behavior, a bound on
$\left(\delta Y_{i}(t)/Y_{i}\right)$ at an angular frequency $\nu$ should be
interpreted as a bound on the couplings of a DM candidate with a mass of
$m_{\phi}=\frac{\hbar\nu}{2c^{2}}$. Moreover, since the time-averaged value of
$\left\langle\cos^{2}\left(m_{\phi}t\right)\right\rangle=\frac{1}{2}$, there
is an additive constant contribution to the fundamental coupling constants.

The EP bounds on the quadratic interactions are of different nature than the
EP bounds on the linear theory. In the quadratic theory, there is no Yukawa
potential generated at tree level. However the time and space dependent
background value of $\left\langle\phi^{2}\right\rangle$, in the presence of a
massive central object, results with an effective long range force. In
Appendix C, we analyze the obtained background value of
$\left\langle\phi^{2}\right\rangle$ and its affect on the acceleration of a
test body, and compare it to the 1-loop quantum corrections of $\phi^{2}$. We
find that in the regime of our interest of ULDM, the quantum corrections are
negligible compared to the classical ones. See Banks:2020gpu  for a discussion
of the Yukawa force for generalised potentials. Below, we take a closer look
at the profile of $\phi$ and its implications to EP and DDM bounds.

####  1.2.1 The profile of quadratically coupled DM, boundary condition
dependence

The dynamics related to quadratic coupling have a strong dependence on the
boundary conditions. In our analysis, we assume that far away from the Earth,
$\phi$ takes its galactic DM background, following Hees:2018fpg . This
assumption is somewhat idealistic since the Earth is moving in the solar
system, which consists of the moon, other planets, and the Sun, in addition to
our solar system moving within the galactic medium, all perturbing the value
of the scalar. In that way, it might not be entirely realistic to set the
value of $\phi$ to its vacuum solution, assuming it is only be affected by the
Earth, as this may depend on other variables such as the relaxation time and
the dynamical history of the formation of the scalar background Bar:2018acw ;
Budnik:2020nwz ; Balkin:2021zfd ; Bar:2021kti .

Alternatively, one can assume other boundary conditions and consider the
phenomenology of a transient scalar background Dailey_2020 . In fact the
affect of gravitational focusing of ultralight DM in the solar system was
recently analyzed in PhysRevD.105.063032  where it was shown that it leads to
some changes to the distribution of the ultralight DM, distribution that would
have a preferred direction due to the velocity of the Sun in the DM galactic
halo. Finally, the self interaction of the DM in the presence of a central
gravitational potential, is expected to modify the property of the ULDM
distribution, but requires a dedicated focused study beyond the scope of this
paper and will be reported elsewhere haloformation . For the sake of
concreteness and despite the fact that the analysis might be incomplete, we
shall follow the treatment of Hees:2018fpg , assuming trivial DM background at
infinity as in Eq. (1.1).

These boundary conditions lead to two important implications; the first is
that the field is mediating long-distance forces despite being massive, and
the second is a screening behavior near the surface of a massive source. Given
the boundary conditions discussed above, the solution to the Equation of
Motion (EOM) extracted from Eq. (1.8) near a massive central object yields the
following analytical solution to $\phi$

$\phi\left(t,x\right)=\phi_{0}\cos\left(m_{\phi}t+\delta\right)\left[1-s_{C}^{\left(2\right)}[d_{i}^{(2)}]\,\frac{GM_{C}}{r}\right]\,,$
(1.10)

where $\phi_{0}\equiv\sqrt{2\rho_{\rm DM}}/m_{\phi}$ is the background
amplitude of $\phi$ at infinity, $s^{(2)}_{C}[d^{(2)}_{i}]$ is some function
of the quadratic couplings (explained in Appendix A), $G$ is Newton’s
gravitational constant and $M_{C}$ the mass the central body, see subsection B
for additional details.

In this Section, we follow two limits: the weakly coupled limit and the
strongly coupled limit. In those limits, the solution takes the following form

$\phi\left(t,x\right)=\phi_{0}\cos\left(m_{\phi}t+\delta\right)\begin{cases}\left[1-\frac{R_{C}}{r}\left(1-\sqrt{\frac{d_{i}^{\rm
crit}}{d^{(2)}_{i}}}\right)\right]\,{\rm for}\,\,\,d^{(2)}_{i}\gg d_{i}^{\rm
crit}\\\ \left[1-\frac{R_{C}}{r}\frac{d^{(2)}_{i}}{3\,d_{i}^{\rm
crit}}\right]\,\,{\rm for}\,\,\,d^{(2)}_{i}\ll d_{i}^{\rm crit}\,,\end{cases}$
(1.11)

where the critical value of the coupling is defined as $d_{i}^{\rm
crit}=R_{C}/(3Q^{C}_{i}GM_{C})$, where $R_{C}$ is the radius of the central
body. Therefore, for a given DM mass $m_{\phi}$, there exists a critical value
of the quadratic coupling $d^{(2)}$, at which the background value of
$\left\langle\phi^{2}\right\rangle$ is screened, and thus the sensitivity to
quadratically coupled ULDM is suppressed. This can be seen by taking the limit
$r\rightarrow R_{C}$ in Eq.(1.11), when the DDM sensitivity becomes

$\displaystyle\eta^{\rm DDM}\left(\omega\right)$
$\displaystyle=\mathcal{F}_{\omega}\left(\left(\frac{\delta
Y_{i}}{Y_{i}}(t)\right)^{\text{DDM-quad}}_{d^{(2)}_{i}\gg d_{i}^{\rm
crit},r\simeq R_{c}}\right)\simeq\;\frac{\Delta\kappa
d^{(2)}_{i}}{2M_{\text{Pl}}^{2}}\mathcal{F}_{\omega}\left(\left\langle\phi^{2}\right\rangle\right)$
$\displaystyle\simeq\Delta\kappa\frac{\phi_{0}^{2}}{M_{\text{Pl}}^{2}}\,\left[d_{i}^{\rm
crit}+\frac{\Delta r}{R_{C}}d_{i}^{(2)}\sqrt{\frac{d_{i}^{\rm
crit}}{d^{(2)}_{i}}}+\left(\frac{\Delta
r}{R_{C}}\right)^{2}d_{i}^{(2)}\right]\times\mathcal{F}_{\omega}\left(\cos^{2}\left(m_{\phi}t+\delta\right)\right)$
$\displaystyle\simeq\Delta\kappa\frac{\phi_{0}^{2}}{M_{\text{Pl}}^{2}}\,d_{i}^{\rm
crit}\times\mathcal{F}_{\omega}\left(\cos^{2}\left(m_{\phi}t+\delta\right)\right)\,,$
(1.12)

where $\frac{\Delta r}{r}\equiv\frac{r-R_{c}}{r}\ll 1$.

Finally, we comment on the negative coupling scenario (when the sign of the DM
quadratic coupling is opposite relative to the mass term). In this case, the
loss of control over the system is related to the fact that if inside the
Earth the mass is too negative, it would overcome the pressure gradient from
the kinetic term of the order of $1/R_{C}$. Thus, it would lead to tachyonic
instabilities (where within the Earth, the mass squared of the field is
negative) and to a runaway behavior of the field into large-amplitudes inside
and outside Budnik:2020nwz ; Balkin:2021zfd ; Hook:2019pbh .

###  1.3 Summary of EP and DDM sensitivities

We end this introduction by summarizing the sensitivities again, $\eta^{\rm
DDM}$ and $\eta^{\rm EP}$, and provide their full description as a function of
the DM scalar field background. The complete derivation of these sensitivities
can be found in Appendix A. For the linear couplings DM model, we found:

$\displaystyle\eta^{\rm
EP}_{\text{linear}}=2\frac{|\vec{a}_{A,C}-\vec{a}_{B,C}|}{|\vec{a}_{A,C}+\vec{a}_{B,C}|}\simeq\frac{|\Delta
Q_{i}d^{(1)}_{i}\vec{\nabla}V_{C}\left(\phi\right)|}{M_{\text{Pl}}GM_{C}/r^{2}}\simeq\frac{\Delta
Q_{i}d^{(1)}_{i}}{M_{\text{Pl}}GM_{C}/r^{2}}\left[\nabla\phi\left(x,t\right)+\vec{v}\dot{\phi}\left(x,t\right)\right]\,,$
(1.13) $\displaystyle\eta^{\rm
DDM}_{\text{linear}}\left(\omega\right)=\mathcal{F}_{\omega}\left(\frac{\delta
Y(t)}{Y}\right)\simeq\frac{\Delta\kappa_{i}^{A}d^{(1)}_{i}}{M_{\text{Pl}}}\mathcal{F}_{\omega}\left(\left\langle\phi\left(x,t\right)\right\rangle\right)\,.$
(1.14)

In both equation (1.13) and (1.14), the letters A and B represent two
different test bodies, while C denotes the central heavy object such as the
Earth. $d^{(1)}_{i}$ is the linear DM coupling to the $i^{\text{th}}$ SM
field. For the quadratic DM model, we find

$\displaystyle\eta^{\rm
EP}_{\text{quad}}=2\frac{|\vec{a}_{A,C}-\vec{a}_{B,C}|}{|\vec{a}_{A,C}+\vec{a}_{B,C}|}\simeq\frac{\Delta
Q_{i}d^{(2)}_{i}}{M^{2}_{\text{Pl}}GM_{C}/r^{2}}\phi\left(x,t\right)\left[\nabla\phi\left(x,t\right)+\vec{v}\dot{\phi}\left(x,t\right)\right]\,,$
(1.15) $\displaystyle\eta^{\rm
DDM}_{\text{quad}}\left(\omega\right)=\mathcal{F}_{\omega}\left(\frac{\delta
Y(t)}{Y}\right)\simeq\frac{{\Delta\kappa_{i}^{A}d^{(2)}_{i}}}{2M_{\text{Pl}}^{2}}\mathcal{F}_{\omega}\left(\left\langle\phi^{2}\left(x,t\right)\right\rangle\right)\,.$
(1.16)

Finally, we present the approximate sensitivities given the specific DM
background solution of Eq. (1.10) with its special boundary conditions. For
the linear DM model, we get

$\displaystyle\eta^{\rm
EP}_{\text{linear}}=2\frac{|\vec{a}_{A,C}-\vec{a}_{B,C}|}{|\vec{a}_{A,C}+\vec{a}_{B,C}|}\simeq
Q^{C}_{j}d^{(1)}_{j}\Delta Q_{i}d^{(1)}_{i}e^{-m_{\phi}r}\,,$ (1.17)
$\displaystyle\eta^{\rm
DDM}_{\text{linear}}\left(m_{\phi}\right)=\frac{|\delta
Y(t)|}{Y}\simeq\frac{\Delta\kappa_{i}^{A}d^{(1)}_{i}}{M_{\text{Pl}}}\phi_{0}\,.$
(1.18)

Lastly, the results for the quadratic DM model are

$\displaystyle\eta^{\rm
EP}_{\text{quad}}=2\frac{|\vec{a}_{A,C}-\vec{a}_{B,C}|}{|\vec{a}_{A,C}+\vec{a}_{B,C}|}\simeq\Delta
Q_{i}d^{(2)}_{i}s_{C}^{\left(2\right)}[d^{(2)}_{i}]\left(\Delta
Q\right)_{j}^{AB}d^{(2)}_{j}\frac{\phi_{0}^{2}}{2M^{2}_{\text{Pl}}}\left[1-s_{C}^{\left(2\right)}\frac{GM_{C}}{r}\right]\,,$
(1.19) $\displaystyle\eta^{\rm
DDM}_{\text{quad}}\left(2m_{\phi}\right)=\frac{|\delta
Y(t)|}{Y}\simeq\frac{{\Delta\kappa_{i}^{A}d^{(2)}_{i}}}{4M_{\text{Pl}}^{2}}\phi_{0}^{2}\,\left[1-s_{C}^{\left(2\right)}\frac{GM_{C}}{r}\right]^{2}\,.$
(1.20)

##  2 Updated Bounds from EP and DDM searches

In this Section we present the bounds on the DM models both from EP and DDM
searches. We also discuss the interplay between the EP and DDM searches for
the linear and the quadratically coupled DM with the SM. We also discuss two
proposed ways to alleviate the EP test constraints in details and present the
reach of DDM searches in those scenarios.

###  2.1 Summary of current and future bounds

The known bounds on scalar, pseudo-scalar and quadratic DM interactions with
the SM are summarized in Tables 1-3 for various ULDM masses. In addition, we
also present current and future-projected bounds on the linear scalar
couplings $d^{(1)}_{i}$ and on the quadratic couplings $d^{(2)}_{i}$ as a
function of the DM mass for various local DM densities. The bounds for the
electron couplings, $d_{m_{e}}$, are shown in Figure 2, the bounds for the
photon couplings, $d_{e}$, are shown in Figure 3, the bounds for the quark
couplings, $d_{m_{q}}$, are shown in Figure 4, and the bounds for the gluon
couplings, $d_{g}$, are shown in Figure 5. For all linear couplings, the EP
test bounds are derived from the terrestrial Eöt-Wash Be/Ti
Schlamminger:2007ht  and Eöt-Wash Cu/Pb PhysRevD.61.022001  measurements, as
well as from the MICROSCOPE data PhysRevLett.129.121102  taken on a satellite
orbiting the Earth at an approximate altitude of 700 km. For the quadratic
couplings, we present only the bounds from MICROSCOPE, which are expected to
be the strongest Hees:2018fpg . The current DDM bounds for $d_{m_{e}}$ are
given from the H/Si clock-cavity comparison measurements presented in
Kennedy:2020bac . For $d_{e}$, the current DDM bounds are given both from H/Si
and Sr/Si clock-cavity comparisons Kennedy:2020bac , where for masses larger
than $\sim 10^{-16}$ eV an additional measurement with using dynamical
decoupling was applied to improve the sensitivity at high frequencies
Aharony:2019iad . For both $d_{m_{e}}$ and $d_{e}$, we also show a line
representing a DDM sensitivity of $\eta_{\rm DDM}=10^{-18}$ at all masses, as
well as the expected bound from the future DDM MAGIS-100 experiment Abe_2021 .
A DDM experiment involving hyperfine transitions Kennedy:2020bac ;
Hees:2016gop  and/or vibrational levels of a molecule Oswald:2021vtc  can be
used to constraint DM couplings to nucleus i.e. $d_{m_{q}}$ and $d_{g}$.
However, here we present the expected bounds from a nuclear clock with a
sensitivity of $\eta_{\rm DDM}=10^{-24}$, using a Ramsey sequence with the
parameters given in Banerjee:2020kww . The astrophysical constraints mentioned
in Tables 1-3, are coming from various stellar cooling processes as mentioned
in the given references.

$\boxed{m_{\phi}=10^{-8}\,\,{\rm eV}}$

operator | current bound | type of experiment
---|---|---
$\frac{d^{(1)}_{e}}{4\,M_{\text{Pl}}}\phi\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(1)}_{e}\lesssim 10^{-2}$ PhysRevD.61.022001  |  EP test: Eöt-Wash Cu/Pb
$\frac{\tilde{d}^{(1)}_{e}}{M_{\text{Pl}}}\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{e}\lesssim 2\times 10^{8}$ Anastassopoulos:2017ftl  | axion/ALP searches: CAST
$\frac{\left|d^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(1)}_{m_{e}}\right|\lesssim 1$ PhysRevD.61.022001  |  EP test: Eöt-Wash Cu/Pb
$i\frac{\left|\tilde{d}^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{e}}\right|\lesssim 8\times 10^{8}$ Capozzi:2020cbu  | Astrophysics
$\frac{d^{(1)}_{g}\beta(g)}{2M_{\text{Pl}}\,g}\phi\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(1)}_{g}\lesssim 10^{-3}$ Schlamminger:2007ht  |  EP test: Eöt-Wash Cu/Pb
$\frac{\tilde{d}^{(1)}_{g}}{M_{\text{Pl}}}\phi\,G^{\mu\nu}\tilde{G}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{g}\lesssim 10^{8}$ Raffelt:2006cw ; Buschmann:2021juv  | Astrophysics
$\frac{\left|d^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(1)}_{m_{N}}\right|\lesssim 6\times 10^{-3}$ Schlamminger:2007ht  |  EP test: Eöt-wash 2008
$i\frac{\left|\tilde{d}^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{N}}\right|\lesssim 3\times 10^{8}$ Beznogov:2018fda  | Astrophysics
$\frac{d^{(2)}_{e}}{8M^{2}_{\text{Pl}}}\phi^{2}\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(2)}_{e}\lesssim 10^{25}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\left|d^{(2)}_{m_{e}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(2)}_{m_{e}}\right|\lesssim 10^{27}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{d^{(2)}_{g}\beta_{g}}{4M_{\rm Pl}^{2}\,\,g}\phi^{2}\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(2)}_{g}\lesssim 10^{25}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\left|d^{(2)}_{m_{N}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(2)}_{m_{N}}\right|\lesssim 10^{25}$ Berge:2017ovy  |  EP test: MICROSCOPE

Table 1: Strongest existing bounds on various DM couplings for a mass of the
order of $m_{\phi}=10^{-8}\,\,{\rm eV}$.

$\boxed{m_{\phi}=10^{-15}\,\,{\rm eV}}$

operator | current bound | type of experiment
---|---|---
$\frac{d^{(1)}_{e}}{4\,M_{\text{Pl}}}\phi\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(1)}_{e}\lesssim 10^{-4}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\tilde{d}^{(1)}_{e}}{M_{\text{Pl}}}\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{e}\lesssim 2\times 10^{6}$ Reynolds:2019uqt  | Astrophysics
$\frac{\left|d^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(1)}_{m_{e}}\right|\lesssim 10^{-3}$ Berge:2017ovy  |  EP test: MICROSCOPE
$i\frac{\left|\tilde{d}^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{e}}\right|\lesssim 8\times 10^{8}$ Capozzi:2020cbu  | Astrophysics
$\frac{d^{(1)}_{g}\beta(g)}{2M_{\text{Pl}}\,g}\phi\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(1)}_{g}\lesssim 6\times 10^{-6}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\tilde{d}^{(1)}_{g}}{M_{\text{Pl}}}\phi\,G^{\mu\nu}\tilde{G}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{g}\lesssim 10^{8}$ Raffelt:2006cw ; Buschmann:2021juv  | SN1987A, NS
$\frac{\left|d^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(1)}_{m_{N}}\right|\lesssim 6\times 10^{-6}$ Berge:2017ovy  |  EP test: MICROSCOPE
$i\frac{\left|\tilde{d}^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{N}}\right|\lesssim 3\times 10^{8}$ Beznogov:2018fda  | Astrophysics
$\frac{d^{(2)}_{e}}{8M^{2}_{\text{Pl}}}\phi^{2}\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(2)}_{e}\lesssim 10^{11}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\left|d^{(2)}_{m_{e}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(2)}_{m_{e}}\right|\lesssim 10^{12}\,$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{d^{(2)}_{g}\beta_{g}}{4M_{\rm Pl}^{2}\,\,g}\phi^{2}\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(2)}_{g}\lesssim 10^{11}$ Berge:2017ovy  |  EP test: MICROSCOPE.
$\frac{\left|d^{(2)}_{m_{N}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(2)}_{m_{N}}\right|\lesssim 10^{11}$ Berge:2017ovy  |  EP test: MICROSCOPE

Table 2: Strongest existing bounds on various DM couplings for a mass of the
order of $m_{\phi}=10^{-15}\,\,{\rm eV}$.

$\boxed{m_{\phi}=10^{-18}\,\,{\rm eV}}$

operator | current bound | type of experiment
---|---|---
$\frac{d^{(1)}_{e}}{4\,M_{\text{Pl}}}\phi\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(1)}_{e}\lesssim 10^{-4}\,$ Kennedy:2020bac  |  DDM oscillations
$\frac{\tilde{d}^{(1)}_{e}}{M_{\text{Pl}}}\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{e}\lesssim 2\times 10^{6}$ Reynolds:2019uqt  | Astrophysics
$\frac{\left|d^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(1)}_{m_{e}}\right|\lesssim 2\times 10^{-3}$ Kennedy:2020bac  |  DDM Oscillations
$i\frac{\left|\tilde{d}^{(1)}_{m_{e}}\right|}{M_{\text{Pl}}}\phi\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{e}}\right|\lesssim 7\times 10^{8}$ Capozzi:2020cbu  | Astrophysics
$\frac{d^{(1)}_{g}\beta(g)}{2M_{\text{Pl}}\,g}\phi\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(1)}_{g}\lesssim 6\times 10^{-6}$ Berge:2017ovy  |  EP test: MICROSCOPE
$\frac{\tilde{d}^{(1)}_{g}}{M_{\text{Pl}}}\phi\,G^{\mu\nu}\tilde{G}_{\mu\nu}$ |  $\tilde{d}^{(1)}_{g}\lesssim 4$ Abel:2017rtm  | Oscillating neutron EDM
$\frac{\left|d^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(1)}_{m_{N}}\right|\lesssim 2\times 10^{-6}$ Berge:2017ovy  |  EP test: MICROSCOPE
$i\frac{\left|\tilde{d}^{(1)}_{m_{N}}\right|}{M_{\text{Pl}}}\phi\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|\tilde{d}^{(1)}_{m_{N}}\right|\lesssim 4$ Abel:2017rtm  | Oscillating neutron EDM
$\frac{d^{(2)}_{e}}{8M^{2}_{\text{Pl}}}\phi^{2}\,F^{\mu\nu}F_{\mu\nu}$ |  $d^{(2)}_{e}\lesssim 10^{6}$ Kennedy:2020bac  |  DDM oscillations
$\frac{\left|d^{(2)}_{m_{e}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{e}\psi_{e}\psi_{e}^{c}$ |  $\left|d^{(2)}_{m_{e}}\right|\lesssim 10^{6}\,$ Kennedy:2020bac  |  DDM oscillations
$\frac{d^{(2)}_{g}\beta_{g}}{4M_{\rm Pl}^{2}\,\,g}\phi^{2}\,G^{\mu\nu}G_{\mu\nu}$ |  $d^{(2)}_{g}\lesssim 10^{7}$ Berge:2017ovy  |  EP test: MICROSCOPE.
$\frac{\left|d^{(2)}_{m_{N}}\right|}{2M^{2}_{\text{Pl}}}\phi^{2}\,m_{N}\psi_{N}\psi_{N}^{c}$ |  $\left|d^{(2)}_{m_{N}}\right|\lesssim 10^{7}$ Berge:2017ovy  |  EP test: MICROSCOPE

Table 3: Strongest existing bounds on various DM couplings for a mass of the
order of $m_{\phi}=10^{-18}\,\,{\rm eV}$.

Figure 2: Top: the bounds on $d_{m_{e}}^{(1)}$ from the linear DM electrons
couplings. Bottom: the bounds on $d_{m_{e}}^{(2)}$ from the quadratic DM
electrons couplings.

Figure 3: Top: the bounds on $d_{e}^{(1)}$ from the linear DM photons
couplings. Bottom: the bounds on $d_{e}^{(2)}$ from the quadratic DM photons
couplings. Note the bounds in Kennedy:2020bac  were modified for the
assumption of a stochastic DM background.

Figure 4: Top: the bounds on $d_{m_{q}}^{(1)}$ from the linear DM photons
couplings. Bottom: the bounds on $d_{m_{q}}^{(2)}$ from the quadratic DM
photons couplings.

Figure 5: Top: the bounds on $d_{g}^{(1)}$ from the linear DM photons
couplings. Bottom: the bounds on $d_{g}^{(2)}$ from the quadratic DM photons
couplings.

###  2.2 Complementarity of EP tests and DDM searches

We can compare the bounds on the DM couplings coming from DDM experiments to
the ones coming from EP tests, both for linear and quadratic interactions. We
begin by summarizing the scaling of the DDM and EP bounds, both in the linear
and quadratic theories, as presented in Table 4.

| linear theory: $\frac{d_{i}^{(1)}}{M_{\text{Pl}}}\phi\,\mathcal{O}^{i}_{\rm SM}$ | quadratic theory: $\frac{d_{i}^{(2)}}{2M_{\text{Pl}}^{2}}\phi^{2}\,\mathcal{O}^{i}_{\rm SM}$
---|---|---
DDM bounds: $\left(\frac{\delta Y_{i}}{Y}(t)\right)<\eta_{\text{DDM}}$ | $\frac{d_{i}^{(1)}}{M_{\text{Pl}}}\frac{\sqrt{2\rho_{\text{DM}}}}{m_{\phi}}$ | $\frac{d_{i}^{(2)}}{2M_{\text{Pl}}^{2}}\frac{\rho_{\text{DM}}}{m_{\phi}^{2}}$
EP bounds: $\left(\frac{\delta a_{\text{test}}}{a}\right)<\eta_{\text{EP}}$ | $\left(d_{i}^{(1)}d_{j}^{(1)}\right)\Delta Q_{i}^{\text{test}}Q_{j}^{\text{Earth}}$ | $\frac{1}{M_{\text{Pl}}^{2}}\left(d_{i}^{(2)}d_{j}^{(2)}\right)\Delta Q_{i}^{\text{test}}Q_{j}^{\text{Earth}}\,\frac{\rho_{\text{DM}}}{m_{\phi}^{2}}$
Ratio: $\frac{\left(d\right)_{\text{DDM}}}{\left(d\right)_{\text{EP}}}$ | $\frac{M_{\text{Pl}}m_{\phi}\sqrt{\Delta Q_{i}^{\text{test}}Q_{i}^{\text{Earth}}}}{\sqrt{2\rho_{\text{DM}}}}\,\frac{\eta_{\text{DDM}}}{\sqrt{\eta_{\text{EP}}}}$ | $\frac{2M_{\text{Pl}}m_{\phi}\sqrt{\Delta Q_{i}^{\text{test}}Q_{i}^{\text{Earth}}}}{\sqrt{\rho_{\text{DM}}}}\,\frac{\eta_{\text{DDM}}}{\sqrt{\eta_{\text{EP}}}}$
Table 4: Theoretical estimation of the bounds on the linear and quadratic
couplings between the light scalar DM and the SM light fields.
$\eta_{\text{DDM}}$ and $\eta_{\text{EP}}$ are defined as the sensitivity of
the DDM and EP experiments, respectively. $Y$ is a fundamental constant
$Y\in(\alpha,\alpha_{s},m_{f})$ and $\mathcal{O}_{\rm SM}$ is the appropriate
SM operator $\mathcal{O}_{\rm
SM}\in\left(\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\,,\frac{\beta(g)}{2g}G_{a}^{\mu\nu}G^{a}_{\mu\nu}\,,m_{i}\psi_{i}\psi_{i}^{c}\right)$.
The spatial dependence of the bounds is disregarded (equivalent to taking the
$m_{\phi}\ll 1/R_{C}$ limit for the linear case and the $d^{(2)}_{i}\ll
d_{i}^{\rm crit}$ limit for the quadratic case).

As one can easily read from Table 4, the ratio of the bounded couplings from
different types of experiments, i.e., the ratio:
$\left(d\right)_{\text{DDM}}/\left(d\right)_{\text{EP}}$ has the same
parametric dependence in both the linear and the quadratic theory, as long as
the spatial dependence of the bounds may be neglected (namely, away from the
Yukawa decoupling of the linear bounds and in the sub-critical region for the
quadratic bounds). Therefore, under these conditions, if one type of
experiment dominates the bounds on the linear interaction in some region of
the parameter space, we expect it to also dominate the bounds on the quadratic
couplings and vice versa. As is further shown in the table, in agreement with
the plots above, DDM experiments tend to be more powerful at lower masses.
Their corresponding constraints improve linearly with the experimental
sensitivity, whereas EP tests are expected to take over at higher masses while
scaling only with the square root of the experimental sensitivity.

While one of these searches usually dominates the bounds for specific masses,
we would like to argue that EP-tests and the DDM searches are complementary to
each other and provide independent information. Below we point out two
engaging scenarios in which the naive ratio between EP and DDM bounds is
violated, demonstrating their complementary.

####  2.2.1 Enhanced DM Density

The current EP bounds for the quadratic theory and the DDM bounds for both the
linear and the quadratic couplings strongly depend on the local DM density.
These bounds become more stringent if the on-Earth DM density is enhanced
compared to the DM density at the solar position $\rho_{\text{DM}}^{\odot}$,
as would be the case if a compact boson star consisting of $\phi$ is formed in
the early universe, and is gravitationally bounded to the Sun or the Earth
Banerjee:2019epw ; Banerjee:2019xuy . Importantly, note that DDM searches are
more sensitive to the local DM density than EP tests, and thus the ratio of
their corresponding bounds $d_{i}^{\text{DDM}}/d_{i}^{\text{EP}}$ would vary
with the density. The ratios $d_{i}^{\text{DDM}}/d_{i}^{\text{EP}}$, are
presented as a function of the DM on-Earth density enhancement
$\rho_{\text{DM}}/\rho_{\text{DM}}^{\odot}$ for a few different benchmark DM
masses in Figs. 6-7. For
$\left(d_{m_{e}}\right)^{\text{DDM}}/\left(d_{m_{e}}\right)^{\text{EP}}$ and
$\left(d_{e}\right)^{\text{DDM}}/\left(d_{e}\right)^{\text{EP}}$ in Fig 6, the
atomic/molecular clock sensitivity is taken to be $\eta_{\text{DDM}}=10^{-18}$
for all values of $m_{\phi}$. For
$\left(d_{m_{q}}\right)^{\text{DDM}}/\left(d_{m_{q}}\right)^{\text{EP}}$ and
$\left(d_{g}\right)^{\text{DDM}}/\left(d_{g}\right)^{\text{EP}}$ in Fig 7, the
nuclear clock sensitivity is taken to be $\eta_{\text{DDM}}=10^{-24}$ for all
values of $m_{\phi}$. The EP sensitivity is taken from current experiments and
depends on the mass of the DM.

As expected, an enhanced DM density would make the ratio between the DDM
bounds and the EP bounds smaller. In particular, for the electron coupling and
for the photon coupling, the hierarchy between the two searches may be flipped
for masses greater than $\sim 10^{-15}\,{\rm eV}$. In addition, for the
quadratic interactions, the DM density enhancement could also effectively
shift the onset of the critical behavior to higher masses, making DDM searches
sensitive to the quadratic couplings at these masses, as opposed to the
$\rho_{\text{DM}}=\rho_{\text{DM}}^{\odot}$ case. Therefore, when considering
the possibility of a larger DM density, the DDM and EP searches may have
competing sensitivities, making them complimentary.

Figure 6: Ratio between DDM bounds and EP bounds on the DM-SM electron
coupling $d_{m_{e}}$ (top) and photon coupling $d_{e}$ (bottom) as a function
of the on-Earth DM density enhancement, relative to the DM density at the
solar position $\rho_{\text{DM}}^{\odot}$. Solid – ratio for the linear
coupling $d^{(1)}$, dashed – ratio for the quadratic coupling $d^{(2)}$. The
vertical lines mark the minimal density enhancement required to probe sub-
critical quadratic couplings by DDM tests. The DDM sensitivity is taken to be
$\eta_{\text{DDM}}=10^{-18}$ for all DM masses $m_{\phi}$. The EP bound is
taken as from existing EP-tests results.

Figure 7: Ratio between DDM bounds and EP bounds on the DM-SM quark coupling
$d_{m_{q}}$ (top) and the gluon coupling $d_{g}$ (bottom) as a function of the
on-Earth DM density enhancement, relative to the DM density at the solar
position $\rho_{\text{DM}}^{\odot}$. Solid – ratio for the linear coupling
$d^{(1)}$, dashed – ratio for the quadratic coupling $d^{(2)}$. The vertical
lines mark the minimal density enhancement required to probe sub-critical
quadratic couplings by DDM tests. The DDM sensitivity is taken to be
$\eta_{\text{DDM}}=10^{-24}$ for all DM masses $m_{\phi}$. The EP bound is
taken as from existing EP-tests results.

####  2.2.2 Non-generic couplings

Let us discuss the bounds from the EP test experiments, which are generically
stronger than the constraints arising from the DDM searches for individual
coupling in the region $10^{-18}\,{\rm eV}\lesssim
m_{\phi}\lesssim\mathcal{O}(\,{\rm eV})$ Kennedy:2020bac ; Hees:2018fpg . As
discussed, the EP tests compare the dilatonic charges of two test bodies. To
calculate the “dilatonic charge" of an atom a with $Z$($N$) being the number
of protons (neutrons), one can write the mass of an atom $m^{\bf a}$ as,
$m^{\bf a}(Z,N)=m^{\bf a}_{\rm nuc}(Z,N)+Zm_{e}\,,$ where, $m^{\bf a}_{\rm
nuc}$ is the mass of the nucleus of a. Furthermore, the nucleus mass
contribution can be decomposed in terms of the proton ($m_{p}$) and the
neutron ($m_{n}$) masses, and the binding energy of the strong ($E_{3}$) and
electromagnetic ($E_{1}$) interaction as, $m^{\bf a}_{\rm
nuc}(Z,N)=Zm_{p}+Nm_{n}+E_{3}+E_{1}\,.$ Note that $E_{1}$ is dominated by the
EM force within the nucleus, and thus we will ignore the electrons’ effect on
it Damour:2010rp . For a generic atom a, the dilatonic charges, $\vec{Q}^{\bf
a}$, can be written as Damour:2010rp ,

$\\!\\!\vec{Q}^{\mathbf{a}}\\!\approx\\!F^{\mathbf{a}}\\!\\!\left(\\!3\,10^{-4}-4\,r_{I}+8\,r_{Z}\,,\,3\,10^{-4}-3\,r_{I}\,,\,0.9\,,0.09-\frac{0.04}{A^{1/3}}-2\times
10^{6}r_{I}^{2}-r_{Z},0.002\,r_{I}\\!\right)\\!.$

In what follows we use the following notation for a vector $\vec{X}\equiv
X_{e,m_{e},g,\hat{m},\delta m}$ , with $\hat{m}\equiv(m_{d}+m_{u})/2$ ,
$\delta m\equiv(m_{d}-m_{u}),$ $10^{4}\,r_{I;Z}\equiv
1-2Z/A;Z(Z-1)/A^{4/3}\,,$ and
$F^{\mathbf{a}}=931\,A^{\mathbf{a}}/(m^{\mathbf{a}}/{\rm MeV})$ with
$A^{\mathbf{a}}$ being the atomic number of the atom a. The MICROSCOPE
experiment Touboul:2017grn ; Berge:2017ovy , which provides the strongest EP
bounds for masses below $10^{-12}\,$eV, is sensitive to the difference between
the dilatonic charges of Platinum/Rhodium alloy (90/10) and
Titanium/Aluminum/Vanadium (90/6/4) which is given by

$(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta Q\ }})^{\rm Mic}\simeq
10^{-3}(-1.94\,,\,0.03\,,\,0.8\,,\,-2.61\,,\,-0.19)\,.$

$(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta Q\ \\!}})$ for other
experiments looking for EP violation are discussed in Oswald:2021vtc . We find
that these sensitivities map to directions in the five-dimensional ULDM
coupling space that are very different from that of DDM searches, denoted by
$(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta\kappa\ }})^{\rm DDM}$,
which usually have $\mathcal{O}(1)$ sensitives for variations in $\alpha$ and
$m_{e}$ (e.g. Safronova:2019lex ; Antypas:2019yvv ; Antypas:2020rtg  and refs.
therein). Examining the four best EP bounds, Cu-Pb PhysRevD.61.022001 , Be-Ti
Schlamminger:2007ht , and Be-Al Wagner:2012ui  along with the previously
mentioned MICROSCOPE experiment, in the five-dimensional vector space of
coupling, we can construct a combination that would be orthogonal to all of
them, approximately given by,

$\displaystyle\hat{Q}^{\perp}\sim\big{(}0.01\,,\,0.99\,,\,0.01\,,\,-0.01\,,\,0.13\,\big{)}\,.$
(2.1)

It implies that models of light scalar DM with coupling direction defined
according to $\hat{Q}^{\perp}\cdot\vec{d}$ would not be subject to these four
leading EP bounds.

Here for simplicity, we consider an ULDM quadratically coupled to the SM and
assume two scenarios:

  1. 1.

A model where only $d^{(2)}_{m_{e}}\neq 0$,

  2. 2.

A model defined by a vector of sensitivities,
$\hat{Q}^{\perp}\cdot\,\vec{d}^{(2)}$, that is orthogonal to the sensitivities
of the four leading EP test experiments.

We present the bounds on these two models in Figure 8 by the solid and the
dashed lines, respectively. For the second model, we have projected the bounds
on $d^{(2)}_{m_{e}}$ as $\hat{Q}^{\perp}$ has a relatively significant overlap
with the direction corresponding to $m_{e}$ (the second entry of it, which is
the largest). Also for simplicity we have considered a DDM search experiment,
which only depends on $m_{e}$ and thus the sensitivity vector can be written
as $(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta\kappa\ }})^{\rm
DDM}=(\,0,1,0,0,0)\,$. Note that, due to the large overlap of
$\hat{Q}^{\perp}$ with the $m_{e}$ direction, the specific choice of
$(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta\kappa\ }})^{\rm DDM}$ has a
negligible effect on the final conclusion. In our case,
$\hat{Q}^{\perp}\cdot(\stackrel{{\scriptstyle\longrightarrow}}{{\Delta\kappa\
}})^{\rm DDM}\simeq 0.99$, which is approximately the sensitivity coefficients
corresponding to $m_{e}$. In addition, the sensitivity of both the EP tests
and the DDM searches depend on the geometry of the source body as discussed in
Eqs. (1.19) and (1.20) respectively. We also assume a homogeneous spherically
symmetric Earth as the source body, which is made of $32\%$ Iron and $68\%$
silicon oxide. With the above assumption, we get the dilatonic charge of the
source body as

$\displaystyle Q^{\rm source}\simeq
10^{-3}\,(\,1.87\,,\,0.27\,,\,1000.19\,,\,80.51\,,\,0.04\,)\,.$ (2.2)

As mentioned below Eq. (B.6), the critical value of a coupling is inversely
proportional to the corresponding dilatonic charge of the source. In Figure 8,
we see that $d_{m_{e}}^{\rm crit}$ (shown by the solid yellow line) is $35$
times larger than the critical value of the second model which is defined by a
vector of sensitivities, $\hat{Q}^{\perp}\cdot\,\vec{d}^{(2)}$ (shown by the
dashed yellow line) as $Q^{\rm source}\cdot\hat{Q}^{\perp}=9.49\times
10^{-3}$, whereas $Q^{\rm source}_{m_{e}}=0.27\times 10^{-3}$. Unlike the
first model, where only $d_{m_{e}}^{(2)}\neq 0$, the second model is not
constrained by the four leading EP experiments. In the $\hat{Q}^{\perp}$
direction, the strongest EP bound is coming from the Be-Cu test Su:1994gu
(the fifth best one), which is more than five orders of magnitude weaker than
the MICROSCOPE Berge:2017ovy  experiment, which provides the most stringent
bound for any model with only one non-zero coupling Hees:2018fpg . The
turquoise and the pink lines in Figure 8 depict the strongest bounds from the
EP tests on the first and the second models, respectively. The black lines
depict the bounds from DDM searches located on the surface of the Earth,
whereas the red and the blue lines represent bounds from DDM searches located
400 km (average altitude of the International Space Station) and 5000 km above
the surface of the Earth (see FOCOS  clock proposal), respectively. See
Section 3.2 for more details. We have assumed the sensitivity of the DDM
searches to be $\eta_{\rm DDM}=10^{-18}$. Note that, below criticality, even
the terrestrial DDM searches provide stronger bounds than the EP tests for
both the scenarios discussed here. These bounds are $\mathcal{O}(10^{5})$
stronger than that of the EP test in the lower mass region. As explained
before, for a given sensitivity, the reach of the space-based DDM searches is
better than the Earth-based ones. Above $m_{\phi}\gtrsim 2\times
10^{-16}\,{\rm eV}$, the bound from the MICROSCOPE experiment is slightly
stronger than that of the space-based DDM searches for the first scenario,
where only $d_{m_{e}}^{(2)}\neq 0$. However, for the second scenario, due to
the considerable overlap of $\hat{Q}^{\perp}$ with the $m_{e}$ direction, the
reach of the DDM searches is not reduced, unlike the EP tests. This allows the
space-based DDM searches to provide the strongest bound even for the higher
masses and above the critical value of the coupling. Around $m_{\phi}\sim
10^{-13}\,{\rm eV}$, the bounds from DDM searches are $\mathcal{O}(10^{9})$
stronger than the best EP bound (coming from the Be-Cu test shown by the
dashed magenta line in Figure 8).

Figure 8: Exclusion plot for $d^{(2)}_{m_{e}}$; the solid lines assume a model
where only $d^{(2)}_{m_{e}}\neq 0$. The dashed lines depict the bounds for a
model defined by a vector of sensitivities,
$\hat{Q}^{\perp}\cdot\,\vec{d}^{(2)}$, that is orthogonal to the sensitivities
of four leading EP test experiments projected onto $d^{(2)}_{m_{e}}$. The
black lines depict the bounds from DDM searches located on the surface of the
Earth, whereas the red and the blue lines represent bounds from DDM searches
located at 400 km and 5000 km above the surface of the Earth, respectively. We
have assumed the sensitivity of the DDM searches is $\eta_{\rm DDM}=10^{-18}$.
The magenta line depicts the strongest bound from the EP tests (which is
coming from MICROSCOPE experiment PhysRevLett.129.121102 ) for only
$d^{(2)}_{m_{e}}\neq 0$ models. The pink line represents the strongest EP
bound for a model defined by a vector of sensitivities
$\hat{Q}^{\perp}\cdot\,\vec{d}^{(2)}$(which is coming from Be-Cu test
Su:1994gu ). The yellow lines represent the critical value of the couplings.

Let us consider a case where the five-dimensional coupling is universal, thus
does not violate EP. As discussed in Oswald:2021vtc , if a scalar-field
coupling to the SM is defined according to $\vec{Q}_{\rm
dil}\cdot\vec{d}^{(i)}$ where $i=1,2$ with $\vec{Q}_{\rm
dil}\simeq(-0.01,1,1,1,0)\,,$ then it will not be subjected to the EP tests
bounds. However, it will still give rise to deviations from the inverse square
law and, thus, will be constrained by fifth-force search experiments. To
simplify our discussion, we will consider the case of a linearly coupled
scalar to the SM; however, our main result also applies to a quadratically
coupled theory.

To briefly see how the EP non-violation works, we know that gravity couples to
the Ricci scalar $R$, and using Einstein’s equation, one can write $R\propto
T^{\mu}_{\mu}$ where $T^{\mu}_{\mu}$ is the trace of the energy-momentum
tensor. Thus, if a scalar-field coupling to the SM is proportional to
$T^{\mu}_{\mu}$, it will not generate any EP violation. This is an idealistic
limit and is realized only in pure dilaton models, where the dilaton $(\phi)$
couplings are precisely given by

$\mathcal{L}\supset\frac{\phi}{f_{\rm dil}}T^{\mu}_{\mu}\,,$ (2.3)

where $f_{\rm dil}$ is the conformal invariance breaking scale
Dymarsky:2013pqa . As discussed before and in Damour:2010rp ; Hees:2018fpg ,
above interaction would induce a Yukawa interaction between two bodies. The
interaction strength can be written as

$\alpha=\frac{1}{\sqrt{4\pi G_{\rm N}}}\frac{\partial\ln
m(\phi)}{\partial\phi}\propto\frac{1}{f_{\rm dil}}\,.$ (2.4)

This shows that the dilaton coupling is universal, and the conformal
invariance breaking scale determines the coupling strength. The differential
acceleration between two test bodies is proportional to the difference between
their Yukawa interaction strength, as shown in Eq. (A), and it vanishes due to
the universality of the dilaton coupling. Thus, a pure dilaton does not
generate EP violation. However, it is easy to see using Eq. (2.4) that in the
presence of a source with mass $m^{\rm s}$, the acceleration of a test body
can be written as

$\vec{a}_{\rm A}=-\hat{r}\frac{G_{\rm N}\,m^{\rm
S}}{r^{2}}\left[1+\frac{2M_{\rm Pl}^{2}}{f_{\rm dil}^{2}}\left(1+m_{\rm
dil}\,r\right)e^{-m_{\rm dil}\,r}\right],$ (2.5)

where $M_{\rm Pl}^{2}=1/(8\pi G_{N})$ and $m_{\rm dil}$ is the mass of the
dilaton. The above equation shows that the presence of a dilaton causes
deviation from $1/r^{2}$ force and thus can be constrained by various
experiments that test for deviations from Newtonian gravity (fifth-force
searches) (see Fischbach:1996eq  and Refs. therein). We are also assuming that
the dilaton would acquire a small mass from a sector other than the SM
Coradeschi:2013gda  in order to be a viable DM candidate Arvanitaki:2014faa .

So far, we have argued a scalar that interacts with the SM as given in Eq.
(2.3) would not violate EP but would give rise to deviations from Newtonian
gravity. Now to get the direction in the five-dimensional coupling space, we
need to write the expression for $T^{\mu}_{\mu}$. Assuming that the SM is
valid up to the scale $f_{\rm dil}$, $T^{\mu}_{\mu}$ can be written as

$T^{\mu}_{\mu}=\frac{\beta(g)}{2g}G^{2}+\frac{\beta(g_{2})}{2g_{2}}W^{2}+\frac{\beta(y)}{2y}B^{2}+(1-\gamma)\sum_{\psi}m_{\psi}\bar{\psi}\psi\,,$
(2.6)

where $g$, $g_{2}$ are the coupling strength of $SU(3)$ and $SU(2)$ gauge
groups, respectively, $y$ represents the hypercharge corresponding to
$U(1)_{y}$, and $G$, $W$ and $B$ are the corresponding gauge fields
respectively. Also, $\psi$ denotes the SM fermions with mass $m_{\psi}$. For
simplicity, in the above formula, we assume that the conformal breaking scale
$f_{\rm dil}$ is far below the Landau pole of $U(1)_{y}$ and above the
electro-weak (EW) scale. As $T^{\mu}_{\mu}$ is invariant under the evolution
of the Renormalization Group (RG) equation (manifested in the above equation),
the dilaton always couples through anomaly matching to the same quantity at
any scale $\mu$. For our purpose, we consider our theory at $\mu=1\,{\rm GeV}$
and $T^{\mu}_{\mu}$ can be expressed as,

$T^{\mu}_{\mu}=\frac{\beta(g_{s})}{2g_{s}}G^{2}+\frac{\beta(e)}{2e}F^{2}+\sum_{\psi}m_{\psi}\bar{\psi}\psi\,.$
(2.7)

In the above equation, we have redefined the fermion masses in terms of their
pole masses. Combining this with Eq. (2.3) and along with our convention of
defining a vector in the five-dimensional coupling space, we can write the
dilaton coupling vector, $\vec{Q}_{\rm dil}$ as

$\vec{Q}_{\rm dil}=(-\frac{2\beta(e)}{e},1,1,1,0)\,,$ (2.8)

where $e$ is the electric charge and $\beta(e)=e^{3}/(12\pi^{2})$. As below
$m_{e}\sim\,{\rm MeV}$, the theory essentially becomes free, we find
$2\beta(e)/e\sim e^{2}/(6\pi^{2})\times 10\sim 0.015$ as $\log(\rm GeV/\rm
MeV)\sim 10$. Thus, we get $\vec{Q}_{\rm dil}=(-0.01,1,1,1,0)\,.$ Thus, we
have argued that a scalar field (the dilaton), whose coupling is defined
according to $\vec{Q}_{\rm dil}\cdot\vec{d}^{(1)}$, will not generate an EP-
violating acceleration, as the direction is indistinguishable from that of
gravity. In Fig. 9 we show the bounds from various fifth-force searches
projected on the $d^{(1)}_{m_{e}}$ and $d^{(1)}_{g}$ directions on a pure
dilaton model by the blue solid line. For comparison, we also show the bounds
from various EP tests on a model which only couples to either $m_{e}$ or gluon
field strength linearly i.e. $\mathcal{L}\supset
d^{(1)}_{m_{e}}\phi\,m_{e}\bar{e}e/M_{\rm Pl}$ or $\mathcal{L}\supset
d^{(1)}_{g}\beta(g)\,\phi\,G^{2}/(2g\,M_{\rm Pl})$ by the turquoise line.
Various constraints on these models are shown in details in Fig. 2 and Fig. 5
respectively.

We, finally, note that if the dilaton couplings are not perfectly aligned with
that obtained from the trace of the energy-momentum tensor, it will generate
EP-violating acceleration as discussed in Taylor:1988nw ; Kaplan:2000hh .

Figure 9: Bounds from various experiments which are looking for EP-violation
and/or deviation from Newtonian gravity (fifth force searches) on
$d^{(1)}_{m_{e}}$ (left) and $d^{(1)}_{g}$ (right). The turquoise lines show
the strongest constraints from various EP violation searches
MICROSCOPE:2022doy ; Wagner:2012ui ; Schlamminger:2007ht ; Smith:1999cr
assuming a model where only $d^{(1)}_{i}\neq 0$ where $i=m_{e},g$. The blue
lines depict the strongest bound coming from various fifth force experiments
(see Fischbach:1996eq ; Lee:2020zjt ; Tan:2020vpf  and Refs. therein). The red
dashed lines indicate $d^{(1)}_{i}=1$.

##  3 Quadratic Interactions and Screening

In the previous section, we observed that the bounds on the quadratic
couplings are weaker than the bounds on the linear couplings. One reason is
the cutoff-suppression, attenuating the effects of the quadratic coupling by a
factor of $\phi/\Lambda_{\rm UV}$ compared to those of the linear coupling,
where $\Lambda_{\rm UV}$ is the UV scale that characterizes the EFT. This
happens as a quadratic couplings represents a higher-dimensional effective
operator than the linear one.

The other reason is the fact that the quadratic coupling might be screened at
the surface of a central body as discussed in Section 1.2.1, as well as in
Appendix A, and previously in Hees:2018fpg . Below we discuss two important
scenarios that alter the effects of the screening behavior. The first is a
theoretical one - a model in which both linear and quadratic couplings are
present simultaneously, and the second is an experimental one - positioning
DDM experiments in space.

###  3.1 Screening and criticality in a model with both linear and quadratic
couplings

Let us discuss the sensitivity of EP tests and DDM searches in the presence of
both linear and quadratic couplings between the ULDM and the SM. As we will
show here, a theory with linear and quadratic couplings has more severely
constrained linear and quadratic couplings than a theory with only one of the
couplings turned on. Also, in the presence of both the couplings, there is a
small region of the parameter space where the DDM bounds are stronger than
those of the EP tests.

The Lagrangian of that model can be written as

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{lin-int}}+\mathcal{L}_{\text{
quad-int}}\,,$ (3.1)

where $\mathcal{L}_{\text{lin-int}}$ and $\mathcal{L}_{\text{ quad-int}}$ are
defined in Eq. (1.2) and Eq. (1.8) respectively. As we are interested in
calculating the sensitivities of DDM searches and EP tests, we would like to
solve for the profile of $\phi$. As discussed above, we assume a homogeneous
boundary condition at large distances for $\phi$ i.e.

$\phi(r\to\infty,t)=\phi_{0}\cos(m_{\phi}t+\delta)\,,$ (3.2)

with $\phi_{0}=\sqrt{2\rho_{\rm DM}}/m_{\phi}$. The EOM of this combined model
is

$\left(\frac{\partial^{2}}{\partial
t^{2}}-\nabla\cdot\nabla+\tilde{m}_{\phi}^{2}\left(r\right)\right)\phi=J_{\text{source}}\left(r\right)\,,$
(3.3)

where we define

$\displaystyle\tilde{m}_{\phi}^{2}\left(r\right)\equiv
m_{\phi}^{2}+\frac{Q_{i}^{C}d_{i}^{(2)}}{M_{\text{Pl}}^{2}}\rho_{C}\left(r\right)\,\,{\rm
and}\,\,\,\,J_{\text{source}}\left(r\right)\equiv-\frac{Q^{C}_{i}d_{i}^{\left(1\right)}}{M_{\text{Pl}}}\rho_{C}\left(r\right).$
(3.4)

Notice that the linear coupling, $d_{i}^{(1)}$, provides a source term for the
EOM whereas the quadratic coupling $d_{i}^{(2)}$ modifies the mass term of
$\phi$. As discussed in Hees:2018fpg , in the presence of a source body $C$,
the SM fields can be replaced by the density of the source body
$\rho_{C}\left(r\right)$ with corresponding dilatonic charge $Q_{i}^{C}$,
where $i$ runs over the SM species coupled to the ULDM. If we model the source
body $C$ as a uniform density sphere of radius $R_{C}$ and mass $M_{C}$, then
one can write

$\rho_{C}\left(r\right)=\begin{cases}\frac{3M_{C}}{4\pi R_{C}^{3}}&r\leq
R_{C}\\\ 0&r>R_{C}\,.\end{cases}$ (3.5)

See the discussion around Eq. (B.1) for more details. The solution of the EOM
given in Eq. (3.3) with the boundary condition of Eq. (3.2) can be written as

$\displaystyle\phi\left(r,t\right)=$ $\displaystyle\begin{cases}\phi_{0}\,{\rm
cs}\,\frac{\text{sinhc}\left(\frac{r}{R_{C}}\sqrt{d_{i}^{(2)}/d_{i}^{\rm
crit}}\right)}{\cosh\left[\sqrt{d_{i}^{(2)}/d_{i}^{\rm
crit}}\right]}-M_{C}\frac{r^{2}}{R_{C}^{3}}\frac{Q^{C}_{j}d_{j}^{\left(1\right)}}{M_{\text{Pl}}}I\left(\tilde{m}_{\phi}r\right)\frac{e^{-\tilde{m}_{\phi}r}}{4\pi}&r<R_{C}\\\
\phi_{0}\,{\rm
cs}\,\left(1-s_{C}[d_{i}^{(2)}]\frac{GM_{C}}{r}\right)-M_{C}\frac{Q^{C}_{j}d_{j}^{\left(1\right)}}{M_{\text{Pl}}}I\left(m_{\phi}R_{c}\right)\frac{e^{-m_{\phi}r}}{4\pi
r}&r\geq R_{C}\,,\end{cases}$ (3.6)

where we define ${\rm cs}=\cos\left(m_{\phi}t+\delta\right)$ and the functions
$I(x)$ and $s_{C}[d_{i}^{(2)}]$ are given by

$\displaystyle I(x)=3\,\dfrac{x\cosh x-\sinh
x}{x^{3}}\,,\,\,s^{(2)}_{C}[d^{(2)}_{i}]=Q^{C}_{i}d^{(2)}_{i}J_{+}\left(\sqrt{d^{(2)}_{i}/d_{i}^{\rm
crit}}\right)\,.$ (3.7)

In addition, $d_{i}^{\rm crit}=R_{C}/(3Q^{C}_{i}GM_{C})$ as defined before,
and the function $J_{+}(x)$ is defined as $J_{+}(x)=3(x-\tanh x)/x^{3}$
Hees:2018fpg . $G$ and $M_{\rm Pl}$ can be used interchangeably with $4\pi
G=1/M_{\rm Pl}^{2}\,$. Using the solution of the EOM for $r\geq R_{C}$, and
assuming the signal will be time averaged, we get the sensitivity of the EP
tests as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\eta^{\text{EP}}(r)$
$\displaystyle=$ $\displaystyle
2\frac{|\vec{a}_{A,C}-\vec{a}_{B,C}|}{|\vec{a}_{A,C}+\vec{a}_{B,C}|}\simeq\frac{\left(\Delta
Q\right)_{i}^{AB}}{GM_{C}/r^{2}}\left(\frac{d^{(1)}_{i}}{M_{\text{Pl}}}+\frac{d^{(2)}_{i}}{M^{2}_{\text{Pl}}}\phi\right)|\vec{\nabla}\phi|$
(3.8) $\displaystyle\simeq$ $\displaystyle\,\left(\Delta
Q\right)_{i}^{AB}\Bigg{(}s_{C}^{\left(2\right)}[d_{j}^{(2)}]d^{(2)}_{i}\frac{\phi_{0}^{2}}{2M^{2}_{\text{Pl}}}\left[1-s_{C}^{\left(2\right)}[d_{j}^{(2)}]\frac{GM_{C}}{r}\right]$
$\displaystyle+Q^{C}_{j}d_{j}^{(1)}d^{(1)}_{i}\,I(m_{\phi}R_{C})\,(1+m_{\phi}r)\,e^{-m_{\phi}r}$
$\displaystyle-\left(Q^{C}_{j}d_{j}^{(1)}\right)^{2}d^{(2)}_{i}\frac{G\,M_{C}}{r}I(m_{\phi}R_{C})^{2}\,(1+m_{\phi}r)\,e^{-2m_{\phi}r}\Bigg{)}\,,$

and the sensitivity of DDM searches as

$\displaystyle\eta^{\text{DDM}}_{\omega}(r)=$ $\displaystyle\;\frac{|\delta
Y(t)|}{Y}\approx\frac{\Delta\kappa_{i}d_{i}^{(2)}}{2M_{\text{Pl}}^{2}}\,\times\phi_{0}^{2}\left(1-s_{C}[d_{j}^{(2)}]\frac{GM_{C}}{r}\right)^{2}\mathcal{F}_{\omega}\left({\rm
cs}^{2}\right)$ $\displaystyle-\;$
$\displaystyle\frac{\Delta\kappa_{i}d_{i}^{(2)}}{M_{\text{Pl}}^{2}}\frac{M_{C}Q^{C}_{j}d_{j}^{\left(1\right)}}{M_{\text{Pl}}}I\left(m_{\phi}R_{c}\right)\frac{e^{-m_{\phi}r}}{4\pi
r}\times\phi_{0}\,\left[1-s_{C}[d_{k}^{(2)}]\frac{GM_{C}}{r}\right]\mathcal{F}_{\omega}\left(\rm
cs\right)$ $\displaystyle+\;$
$\displaystyle\frac{\Delta\kappa_{i}d_{i}^{(1)}}{M_{\text{Pl}}}\,\times\phi_{0}\,\left(1-s^{(2)}_{C}[d_{j}^{(2)}]\frac{GM_{C}}{r}\right)\mathcal{F}_{\omega}\left({\rm
cs}\right)\,.$ (3.9)

We want to describe the screening effect in this model. As most of the DDM
searches are terrestrial, in this section we consider them to be performed
very close to the surface of the source body, i.e., at $r\simeq R_{C}$. In
Section 3.2 we discuss the space based DDM searches where $r\gtrsim R_{C}$.

As discussed before, in a model where the DM interacts only quadratically with
the SM, if the coupling is larger than the critical value, the DDM sensitivity
is screened, and the dependence on the quadratic couplings is suppressed.
However, in a model with both linear and quadratic couplings, due to the mixed
$d^{(2)}d^{(1)}$ term, there is no such criticality for the DDM searches.

To see how it works, let us start with the case when $d^{(2)}_{i}\gg
d_{i}^{\rm crit}$. In this limit $s^{(2)}_{C}[d^{(2)}_{i}]$ can be written as
Eq. (B.6) and we get,

$\displaystyle
1-s^{(2)}_{C}[d_{i}^{(2)}]\frac{GM_{C}}{r}=\sqrt{\frac{d_{i}^{\rm
crit}}{d^{(2)}_{i}}}+\mathcal{O}\left(\frac{\Delta r}{R_{C}}\right)\,,$ (3.10)

and Eq. (3.1) becomes

$\displaystyle\left.\eta^{\text{DDM}}(\omega)\right|_{\left(d^{(2)}_{i}\gg
d_{i}^{\rm crit},\>r\simeq R_{c}\right)}$
$\displaystyle\approx\frac{\Delta\kappa_{i}}{M_{\text{Pl}}^{2}}\phi_{0}^{2}\,\,d_{i}^{\rm
crit}\times\mathcal{F}_{\omega}\left({\rm cs}^{2}\right)$
$\displaystyle-\frac{\Delta\kappa_{i}}{M_{\text{Pl}}}\phi_{0}\,\left[\frac{Q^{C}_{j}}{3Q^{C}_{i}}d_{j}^{(1)}\sqrt{\frac{d_{i}^{(2)}}{d_{i}^{\rm
crit}}}\right]I\left(m_{\phi}R_{c}\right)e^{-m_{\phi}R_{c}}\times\mathcal{F}_{\omega}\left({\rm
cs}\right)$
$\displaystyle+\frac{\Delta\kappa_{j}}{M_{\text{Pl}}}\phi_{0}\,\left[d_{j}^{(1)}\sqrt{\frac{d_{i}^{\rm
crit}}{d_{i}^{(2)}}}\right]\times\mathcal{F}_{\omega}\left({\rm
cs}\right)+\mathcal{O}\left(\frac{\Delta r}{R_{C}}\right)\,.$ (3.11)

As expected, the above equation shows that if we have only quadratic
couplings, i.e., $d_{i}^{(1)}=0$, then DDM searches become insensitive to the
quadratic couplings. However when $d_{i}^{(1)}\neq 0$, it is sensitive to
$\sqrt{d_{i}^{(2)}}$. We can further simplify the above expression for
$m_{\phi}\ll 1/R_{C}$ as

$\displaystyle\\!\\!\left.\eta^{\text{DDM}}\left(\omega\right)\right|_{\left(d^{(2)}_{i}\gg
d_{i}^{\rm crit},\>r\simeq R_{c}\ll
1/m_{\phi}\right)}\approx-\frac{\Delta\kappa_{j}}{3M_{\text{Pl}}}\phi_{0}\,d_{j}^{(1)}\sqrt{\frac{d_{i}^{(2)}}{d_{i}^{\rm
crit}}}\times\mathcal{F}_{\omega}\left({\rm
cs}\right)+\mathcal{O}\left(\frac{\Delta r}{R_{C}},\frac{d_{i}^{\rm
crit}}{d_{i}^{(2)}}\right)\,,$ (3.12)

and for $m_{\phi}\gg 1/R_{C}$ as

$\displaystyle\left.\eta^{\text{DDM}}\right|_{\left(d^{(2)}_{i}\gg d_{i}^{\rm
crit},\>r\simeq R_{c}\gg
1/m_{\phi}\right)}\approx\frac{\Delta\kappa_{j}}{M_{\text{Pl}}}\phi_{0}\,d_{j}^{(1)}\sqrt{\frac{d_{i}^{(2)}}{d_{i}^{\rm
crit}}}\left(\frac{d_{i}^{\rm
crit}}{d_{i}^{(2)}}-\frac{1}{2m_{\phi}^{2}R_{C}^{2}}\right)\times\mathcal{F}_{\omega}\left({\rm
cs}\right)+\mathcal{O}\left(\frac{\Delta r}{R_{C}}\right)\,,$ (3.13)

by noting the limiting case of the function $I(x)$

$\displaystyle\lim_{x\to 0}I(x)\to 1\,\,{\rm
and}\,\,\lim_{x\to\infty}I(x)e^{-x}\to\frac{1}{2x^{2}}\,.$ (3.14)

For completeness we also discuss the case of small quadratic coupling,
$d^{(2)}_{i}\ll d_{i}^{\rm crit}$. In this case $s_{C}[d^{(2)}_{i}]\simeq
Q^{C}_{i}d^{(2)}_{i}$ and Eq. (3.1) becomes

$\displaystyle\left.\eta^{\text{DDM}}\right|_{\left(d^{(2)}_{i}\ll d_{i}^{\rm
crit},\>r\simeq
R_{c}\right)}\simeq\frac{\Delta\kappa_{i}d_{i}^{(2)}}{2M_{\text{Pl}}^{2}}\,\times\phi_{0}^{2}\,{\rm
cs}^{2}+\frac{\Delta\kappa_{i}d_{i}^{(1)}}{M_{\text{Pl}}}\,\times\phi_{0}\,{\rm
cs}+\mathcal{O}\left(\frac{d_{i}^{(2)}}{d_{i}^{\rm crit}}\right)\\!,\\!\\!$
(3.15)

for all masses.

In Figures. 10-11, we present the allowed parameter space of a model with both
the linear and quadratic couplings for different masses of the DM. We notice
that introducing a non-zero quadratic coupling changes the bounds on the
linear coupling and vice-versa. This means that a theory that has both linear
and quadratic couplings has a stronger constraint on each compared to a theory
with only one of the couplings. Despite that, we see that in most of the
parameter space of interest, the EP bounds from the linear coupling are still
the dominant ones. Due to the scalar field profile in the presence of both
linear and quadratic couplings, there is a small region of the parameter space
where the DDM bounds are stronger than the EP bounds, as can be seen in all
the figures below. This could be of potential interest to DDM searches whose
accuracy has improved vastly in the last few years.

The above observation motivates us to present a few models where the linear
couplings are suppressed compared to the quadratic ones. We require that these
theories are natural in the sense that there is no fine-tuning in order to
achieve such hierarchies between the small values of the linear coupling
compared to the relatively large values of the quadratic coupling. We present
two such models within the Clockwork framework in Section 5.4, and another
model of within the Relaxed Relaxion framework in Section 6.2.3.

Figure 10: Allowed parameter space for the linear and quadratic electron
coupling for masses of $m_{\phi}=10^{-13}\,\,{\rm
eV}\,\text{{(top)}},\>10^{-15}\,\,{\rm eV}\,\text{{(bottom)}}$.

Figure 11: Allowed parameter space for the linear and quadratic gluon coupling
for masses of $m_{\phi}=10^{-14}\,\,{\rm
eV}\,\text{{(top)}},\>10^{-15}\,\,{\rm eV}\,\text{{(bottom)}}$.

###  3.2 Screening and criticality in space

Since the screening of the ULDM is most dominant at the surface of the Earth,
experiments done further away are less affected by it, as can be seen by Eq.
(1.11). This is the key to the dominance of the MICROSCOPE EP test, positioned
at an altitude of roughly $700$ km, over the bounds on the quadratic coupling.
If, however, DDM searches are also performed in geocentric orbits, they too
would become sensitive to the ULDM quadratic couplings. To demonstrate this
point, we show in Fig. 12 the bounds on the electron coupling and photon
coupling, $d^{(2)}_{m_{e}}$ and $d^{(2)}_{e}$ respectively, as expected for
DDM experiments with sensitivities of $10^{-18}$, and $10^{-20}$, located at
400 km, 5000 km, and 23000 km above the surface of the Earth. Below we survey
some of the recent proposals with the potential to launch highly sensitive DDM
experiments into space.

The NASA Deep Space Atomic Clock (DSAC) mission has recently demonstrated a
microwave trapped ion clock based on Hg+ ions achieving a factor of $10$
improvement over previous space-based clocks Burt2021 . The such clock was
proposed for the auto-navigation of spacecraft 2018Navigation . Cold atom
microwave clock was demonstrated in space in CAC2018 . The ACES (Atomic Clock
Ensemble in Space) mission ACES  is planned to perform an absolute measurement
of the red-shift effect between the microwave PHARAO clock on-board the
International Space Station (ISS) and clocks on Earth, to improve such limit
by an order of magnitude.

The progress in the development of optical clocks has been extraordinary, with
three orders of magnitude improvement in uncertainty over the last 15 years
LudBoyYe15 . Several optical clocks have reached uncertainty at the 10-18
level (see, e.g., PhysRevLett.123.033201 ), with further improvements expected
as there is no apparent technical limit. Portable high-precision optical
clocks were also demonstrated KelBurKal19 , which is a prerequisite for space
deployment. Various clock-comparisons and clock-cavity comparison experiments
are sensitive to $d_{e}^{(i)}$, $d_{m_{e}}^{(i)}$, and $d_{g}^{(i)}$. The
applications of the different clock types of clocks and optical cavities for
ULDM searches were recently reviewed in 2022UDM .

Deployment of high-precision optical clocks in space will enable both
practical and fundamental applications, including tests of general relativity
FOCOS , DM searches Sun , gravitational wave detection in new wavelength
ranges Kolkowitz_2016 ; Fedderke:2021kuy , relativistic geodesy
Puetzfeld:2019kki , linking Earth optical clocks Gozzard:2021wkf , and others.
The roadmap for cold atom technologies in space has been outlined in roadmap .
In the present work, we demonstrate another window of opportunity to study
ULDM with clocks in space, taking advantage of the space environments that are
drastically different from that of the Earth. Being away from the Earth’s
surface allows us to test the quadratic models described above. We used
orbital parameters of proposal FOCOS  as an example in Section 2.2.2. Ref.
FOCOS  describes a space mission concept that would place a state-of-the-art
optical atomic clock in an eccentric orbit around Earth. The main mission goal
is to test the gravitational red-shift, a classical test of general
relativity, with sensitivity 30,000 times beyond current limits by comparing
clocks on Earth and the spacecraft. A high stability laser link between the
Earthbound and space clocks is needed to connect the relative time, range, and
velocity of the orbiting spacecraft to Earthbound stations. In general
relativity, the tick rate of time slows in the presence of massive bodies, and
locating one or more clocks in orbits around the Earth provides a low-noise
environment for tests of gravity. Sr high-precision optical atomic clock
aboard an Earth-orbiting space station (OACESS) OACESS  was proposed for DM
searches, including static or quasi-static apparent variations of $m_{e}$ with
changing height above Earth’s surface and transient changes in the apparent
value of $\alpha$ due to the passage of a relativistic scalar wave from an
intense burst of extraterrestrial origin. The goal of this pathfinder mission
is to compare the space-based high-precision optical atomic clock (OAC) with
one or more ultra-stable terrestrial OACs to search for space-time-dependent
signatures of dark scalar fields that manifest as anomalies in the relative
frequencies of station-based and ground-based clocks. The OACESS will serve as
a pathfinder for dedicated missions (e.g., FOCOS described above FOCOS ) to
establish high-precision OAC as space-time references in space. We used the
orbital parameters of the International Space Station ($\sim$ 400 km above the
surface of the Earth) and FOCOS mission proposal FOCOS  in Fig. 8 and Fig. 12.

A version of the such proposal with a state-of-the-art cavity will enable test
the quadratic models by also running a clock-cavity experiment as carried out
in Kennedy:2020bac , sensitive to $d_{e}^{(2)}$ coupling. We note that a time
transfer link to Earth is not required for such an experiment. The clock-
comparison experiment in space involving a molecular clock will also be
sensitive to $d_{m_{e}}^{(2)}$. Molecular clocks are projected to reach
$10^{-18}$ uncertainties HanKuzLun21 .

In Ref. Sun , a clock-comparison satellite mission with two clocks onboard to
the inner reaches of the solar system was proposed to search for a DM halo
bound to the Sun and to look for the spatial variation of the fundamental
constants associated with a change in the gravitation potential. Various clock
combinations were considered to provide sensitivities to various couplings.
This work showed that the projected sensitivity of space-based clocks for the
detection of Sun-bound DM halo exceeds the reach of Earth-based clocks by
orders of magnitude. This mission in its proposed form can be used to test the
quadratic coupling models. A DM halo bound to the Sun can drastically improve
the experimental reach due to much higher DM densities.

Figure 12: The bounds on the quadratic interactions with electrons (top) and
photons (bottom) from a hypothetical DDM experiment done on satellites
orbiting at different radii.

##  4 Theoretical Challenges of Models with Quadratic Couplings

In this section, we describe the theoretical issues related to theories with
sizable quadratic couplings. The first is the EFT expectation setting a
hierarchy between the linear and the quadratic couplings of a generic theory.
The second is the naturalness problem caused by the lightness of the scalar,
as a desirable large scalar quadratic coupling is associated with high
corrections to the scalar mass. In Section 5, we present the symmetry
principles that can give rise to a large hierarchy between linear and
quadratic interactions, and in Section 6 we present a model in which in
addition the scalar is kept dynamically light even for detectable quadratic
couplings.

##### Linear vs. quadratic EFT:

Consider any naive dimension 4 (not necessarily of anomalous dimension greater
than 4) SM operator, $\mathcal{O}_{\text{SM}}$. Since the action of the theory
is dimensionless, any coupling between the scalar DM field and this operator
has to be suppressed by some cutoff of the theory, denoted by $\Lambda$. For a
linear coupling, we have only one power of $\Lambda$ suppression,
$\frac{\phi}{\Lambda}\mathcal{O}_{\text{SM}}$, while for a quadratic coupling,
we have two powers of $\Lambda$ suppression,
$\frac{\phi^{2}}{\Lambda^{2}}\mathcal{O}_{\text{SM}}$. In the Wilsonian
picture of RG, $\Lambda$ is expected to be the largest scale of the described
system. As a result, we naively expect a large hierarchy between the linear
and quadratic interaction strength.

##### Naturalness:

Consider the following interaction between the DM field and the SM fermions,

$\mathcal{L}_{\text{int}}=\frac{d^{(2)}_{m_{e}}}{\Lambda^{2}}m_{e}\phi^{2}\psi_{e}\psi_{e}^{c}\,,\qquad\rho_{A}=\psi_{e}\psi_{e}^{c}\
\,,$ (4.1)

where $\psi_{e}$ is the electron field. In a natural model, the quantum
corrections to the mass of the light scalar, $\delta m_{\phi}^{2}$, are small
compared to the classical bare mass parameter, $m_{\phi}^{2}$. The simplest
1-loop diagram that contributes to $\delta m_{\phi}^{2}$ is presented in Fig.
13.

Figure 13: 1-loop correction to the new scalar mass in the
$\frac{d^{(2)}_{m_{\psi}}}{\Lambda^{2}}m_{\psi}\phi^{2}\psi\psi^{c}$ model.

Given the above electron coupling, the diagram gives the following mass
correction

$\delta(m^{2}_{\phi})^{\text{1-loop}}\sim\frac{4}{16\pi^{2}}\times\frac{d^{(2)}_{m_{e}}}{\Lambda^{2}}\left(m_{e}\Lambda^{e}_{\text{UV}}\right)^{2}\,,$
(4.2)

where $m_{e}$ is the electron mass, and $\Lambda^{e}_{\text{UV}}$ is the
effective cut-off of the loop in Fig 13, where new degrees of freedom are
required to cancel the UV sensitivity of the scalar mass. The requirement
$\delta m_{\phi}^{2}\lesssim m_{\phi}^{2}$ yields

$\Lambda^{e}_{\text{UV}}\lesssim\Lambda\,\frac{2\pi}{\sqrt{d^{(2)}_{m_{e}}}}\frac{m_{\phi}}{m_{e}}\,.$
(4.3)

For an ultra-light scalar, a measurable coupling of Eq. (4.1) requires a very
small cutoff of the theory. For example, if we parameterize the experimental
reach in terms of the effective temporal variation of the electron mass
$\delta m_{e}/m_{e}$, induced by the ULDM field $\phi$, we find that

$\Lambda^{e}_{\text{UV}}\lesssim\frac{2\pi}{m_{e}}\sqrt{\frac{2\rho_{\rm
DM}}{\left(\delta m_{e}/m_{e}\right)^{\rm exp}}}\lesssim 20\,\,{\rm
eV}\left(\frac{10^{-18}}{\left(\delta m_{e}/m_{e}\right)^{\rm
exp}}\right)^{1/2}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)^{1/2}\\!,$ (4.4)

where, $\rho_{\text{DM}}^{\odot}=0.4\,{\rm GeV}/(\rm cm)^{3}$ is the mean
galactic DM energy density, similar to the one expected in our solar system
Salucci:2010qr , and $\left(\delta m_{e}/m_{e}\right)^{\rm exp}$ is defined as
the experiment sensitivity to the variations of the electron mass. The
relatively low cutoff of Eq. (4.4) is theoretically unfavorable, as it
suggests that there exist some new fields with masses below $20\,{\rm eV}$
that are coupled to the SM fermions. The same analysis can be done for DM
coupling to the photons, the quarks and the gluons as well. The quarks
couplings yield the following bound

$\Lambda_{\text{UV}}^{\text{q}}\lesssim\frac{2\pi}{m_{q}}\sqrt{\frac{\rho_{\rm
DM}}{\left(\delta m_{q}/m_{q}\right)^{\rm exp}}}\lesssim 1\,\,{\rm
keV}\left(\frac{10^{-22}}{\left(\delta m_{q}/m_{q}\right)^{\rm
exp}}\right)^{1/2}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)^{1/2}\\!.$ (4.5)

Due to the difference in the degree of divergence, the parametric form of the
cut-offs for the DM couplings to the photons and the gluons are different than
that of Eqs. (4.4)-(4.5). For the gluon coupling, we get

$\Lambda_{\text{UV}}^{\text{g}}\lesssim\left(8\pi^{2}\frac{\rho_{\rm
DM}}{\left(\delta\alpha_{s}/\alpha_{s}\right)^{\rm exp}}\right)^{1/4}\lesssim
40\,\,{\rm
keV}\left(\frac{10^{-22}}{\left(\delta\alpha_{s}/\alpha_{s}\right)^{\rm
exp}}\right)^{1/4}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)^{1/4}\\!.$ (4.6)

The cutoff can be raised if the on-Earth DM density, which drives the
oscillations observed in terrestrial DDM searches, is enhanced compared to the
mean galactic DM density.333A much higher density is allowed if the DM is
forming a halo around the Earth Banerjee:2019epw ; Banerjee:2022wzk ;
haloformation . Note that while the parameterisation above can be easily
applied to any DDM experiment probing such oscillations, the effect of a DM
density enhancement on the experimental sensitivity is model-dependent and
experiment-dependent. One should also note that the, existence of new physics
at the keV scale is constrained by various astrophysical and cosmological
considerations, which are subjected to recent critical investigations
Allahverdi:2020bys ; DeRocco:2020xdt . A linear coupling between the DM and
the SM may also possess a naturalness problem for CP-even scalars. However, it
could be protected due to the existence of accidental/non-accidental
symmetries, which are absent in the case of a quadratic coupling. In addition,
a CP-odd linear theory can be embedded in an axion-like theory, thus protected
from a fine-tuning problem, as explained in the following section.

##  5 Examples of Technically-Natural Models of DM with Quadratic Interactions

In this section, we survey a few models which yield a technically-natural
hierarchy between the linear ULDM coupling and the quadratic one, allowing the
quadratic interactions to dominate the phenomenology of the ULDM. We begin by
addressing the symmetries protecting these couplings in the agnostic EFT
approach, identifying those that may retain the linear couplings small in
subsection 5.1. We then specify two models in which the linear interactions
are absent - a pseudo Nambu-Goldstone boson (pNGB) effective model in
subsection 5.2 and a UV-complete Higgs-portal model in subsection 5.3.
Finally, in subsection 5.4 we present two variations of the Clockwork
framework in which the the hierarchy between the linear and the quadratic
couplings can be ameliorated in a technically-natural way. Although these
models present theoretically-sound mechanisms for altering the naive hierarchy
between the strength of the quadratic and the linear interactions, a naturally
light scalar implies the quadratic interactions despite being dominant, are
beyond the reach of current and near future DDM searches. We present a
possible solution to this issue in Section 6.

###  5.1 EFT perspective of the linear vs. the quadratic couplings

In this subsection, we analyze the effective interaction between the $\phi$
sector and the SM without specifying a UV model. We consider the following
three types of operators,

$\mathcal{L}_{\text{int}}\left(\phi\right)=c_{1}^{\text{lin}}\phi\,{\cal
O}_{\text{SM}}+c_{2}^{\text{lin}}\partial_{\mu}\phi\,J_{\text{SM}}^{5\mu}+c_{1}^{\text{quad}}\phi^{2}\,{\cal
O}_{\text{SM}}\,.$ (5.1)

Here we assume $\phi$ is a light pNGB and thus a pseudo-scalar, while ${\cal
O}_{\text{SM}}$ is taken to be a dimension four, CP-even operator, consist of
SM fields only, such as ${\cal
O}_{\text{SM}}=F_{\mu\nu}F^{\mu\nu},\;m_{\psi}\psi\psi^{c}\;$etc,
$J_{\text{SM}}^{5\mu}=\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$ is the SM axial
current. At this point, we ignore possible self coupling of $\phi$ and the
possible interactions with some hidden sectors.

In general, we expect the linear $\phi$ couplings to dominate over the
quadratic couplings unless the quadratic $\phi^{2}$ couplings are protected by
symmetry. In this context, we consider three types of accidental symmetries
that act non-trivially on the field $\phi$

$G_{\text{global}}\supset U(1)_{\Phi}\times CP\times\mathbb{Z}_{2}^{\phi}\,.$
(5.2)

The $U(1)_{\Phi}$ symmetry is a non-linearly realized global $U\left(1\right)$
group, which acts as a constant shift for a pNGB, thus protecting the pNGB
from acquiring a mass. Thus, the smallness of the DM mass $\phi$ is protected
by a shift symmetry,

$U(1)_{\Phi}:\quad\phi\mapsto\phi+\alpha\,.$ (5.3)

Therefore, unless the SM is charged under the $U(1)_{\Phi}$ group, both
$c_{1}^{\text{lin}}\phi\,{\cal O}_{\text{SM}}$ and
$c_{1}^{\text{quad}}\phi^{2}\,{\cal O}_{\text{SM}}$ breaks this symmetry.
Moreover, we consider both $CP$ and $\mathbb{Z}_{2}^{\phi}$ to act similarly
on the field $\phi$. $CP$ is an external symmetry with well-defined
transformation rules for the SM fields, while the internal
$\mathbb{Z}_{2}^{\phi}$ can be taken to affect only $\phi$. For example, we
consider the action of the discrete groups as follows

$\mathbb{Z}_{2}^{\phi}:\quad\phi\mapsto-\phi\,,$ (5.4)

while a general CP transformation in an arbitrary basis can be written as

$\displaystyle CP:\qquad\phi\left(t,\vec{x}\right)$
$\displaystyle\mapsto-\phi\left(t,-\vec{x}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!{\cal
O}_{\text{SM}}\left(t,\vec{x}\right)$ $\displaystyle\mapsto{\cal
O}_{\text{SM}}\left(t,-\vec{x}\right)$ (5.5)
$\displaystyle\\!\\!\\!\\!\\!\\!\partial_{\mu}J^{5\mu}_{\text{SM}}\left(t,\vec{x}\right)$
$\displaystyle\mapsto-\partial_{\mu}J^{5\mu}_{\text{SM}}\left(t,-\vec{x}\right)\,.$
(5.6)

Each operator in Eq. (5.1) may break one or more of the global symmetries. We
summarize the symmetry breaking pattern in Table 5.

| $U\left(1\right)_{\text{shift}}$ | $CP$ | $\mathbb{Z}_{2}^{\phi}$
---|---|---|---
$c_{1}^{\text{lin}}\phi\,{\cal O}_{\text{SM}}$ | x | x | x
$c_{2}^{\text{lin}}\partial_{\mu}\phi\,J_{\text{SM}}^{5\mu}$ | $\checked$ | $\checked$ | x
$c_{1}^{\text{quad}}\phi^{2}\,{\cal O}_{\text{SM}}$ | x | $\checked$ | $\checked$
Table 5: Symmetry preserving and symmetry breaking operators.

In principle, each symmetry can be broken by a different sector. Thus, the
naive expectation that the highly irrelevant operators such as
$\phi^{2}\,{\cal O}_{\text{SM}}$ are less relevant than the linear ones does
not hold. Moreover, the $\mathbb{Z}_{2}^{\phi}$ is considered a good
approximate symmetry if the internal $\mathbb{Z}_{2}^{\phi}$ is highly
protected. Thus, the quadratic coupling i.e.
$c_{1}^{\text{quad}}\phi^{2}\,{\cal O}_{\text{SM}}$, can have the leading
effect on the violation of EP and/or oscillations of the fundamental
constants. The difference between the quadratic theory and a linear is
emphasized through the analytic expressions of the EP constraints, as shown in
Eq. (1.19) and Eq. (1.17). As discussed below Eq. (1.9), a quadratically
coupled DM induces fundamental constants oscillations at an angular frequency
of $\omega=2m_{\phi}$ compare to that of the a linear coupling such as
$c_{1}^{\text{lin}}\phi{\cal O}_{\text{SM}}$
($c_{2}^{\text{lin}}\partial_{\mu}\phi J_{\text{SM}}^{5\mu}$) which yields
fundamental constant oscillations (spin precision) at an angular frequency of
$\omega=m_{\phi}$. In order to be able to probe both the quadratic and the CP-
even linear interaction in DDM experiments, we must require that both the
couplings are not suppressed compared to one another. This criteria posses an
apparent tuning problem from the perspective of a naive EFT analysis. which
can be ameliorated as shown in Subsection 5.4.

###  5.2 A Simple pseudo-Goldstone model as a naturally ultra-light scalar DM
candidate

There are several ways to avoid the naturalness requirement of a small
effective cutoff. One of the most appealing solutions is to consider $\phi$ as
a pNGB of some Spontaneous Symmetry Breaking (SSB). In the non-linear sigma
model description of the Goldstone interactions, one must add an appropriate
linear $\phi$ coupling, as well as other polynomial powers of $\phi$
interactions. The low energy theory of a spontaneous broken $U(1)_{\Phi}$
symmetry can be described by

$\displaystyle\mathcal{L}_{\text{EFT}}=$
$\displaystyle\,\mathcal{L}_{\text{SM}}+\frac{f^{2}}{2}\partial_{\mu}U^{\dagger}\partial^{\mu}U-m_{\psi}U\psi\psi^{c}+\frac{1}{\Lambda^{2}}\partial_{\mu}U^{\dagger}\partial^{\mu}Um_{\psi}U\psi\psi^{c}+...+\text{h.c.}\,,$
(5.7)

where $U=e^{i\frac{\phi}{f}}$, $\psi$ is some SM fermion with mass $m_{\psi}$,
$f$ is the scale at which $U(1)_{\Phi}$ is broken spontaneously,
$\Lambda\gtrsim f$ is the cutoff of the theory, and the ellipsis represents
higher derivatives and/or higher dimensional irrelevant operators in the
Lagrangian. As shown in Eq. (5.14), the last term of Eq. (5.7) gives rise to a
quadratic interaction between the SM fermions and $\phi$. This derivative term
arises naturally from the low-energy physics and does not require an ad-hoc
source in the UV. For example, consider a complex scalar field with a
$U(1)_{\Phi}$ preserving potential

$V(\Phi^{\dagger}\Phi)=\lambda_{\Phi}\left(\Phi^{\dagger}\Phi-\frac{f^{2}}{2}\right)^{2}.$
(5.8)

This potential results in the SSB of the $U(1)_{\Phi}$ symmetry. Therefore, at
low energies, expanding around the true vacuum of the theory, one finds a mass
less Goldstone boson, $\phi$, appearing as the phase of the complex scalar as

$\Phi=\frac{f+\rho}{\sqrt{2}}e^{i\frac{\phi}{f}}\,,~{}$ (5.9)

where $\rho$ is the radial mode of $\Phi$ with mass $m_{\rho}\sim f$. The
above parameterization manifestly provides an interaction between $\rho$ and
$\phi$, which arises from the kinetic term of $\Phi$ as

$\displaystyle\partial^{\mu}\Phi^{\dagger}\partial_{\mu}\Phi$
$\displaystyle\simeq\partial^{\mu}\left(\frac{f+\rho}{\sqrt{2}}e^{-i\frac{\phi}{f}}\right)\partial_{\mu}\left(\frac{f+\rho}{\sqrt{2}}e^{i\frac{\phi}{f}}\right)$
$\displaystyle=\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2f^{2}}\left(f+\rho\right)^{2}\partial_{\mu}\phi\partial^{\mu}\phi\,.$
(5.10)

Thus, Eq. (5.10) suggests that upon integrating out the radial mode, a
coupling between $\rho$ and the SM will result in a low energy effective
coupling between $\partial_{\mu}\phi\partial^{\mu}\phi$ and the SM. Going back
to Eq. (5.7) and expanding the exponent in terms of the small fluctuations of
$\phi/f$, one finds

$\displaystyle\mathcal{L}_{\text{EFT}}\supset$
$\displaystyle\;\mathcal{L}_{\text{SM}}+\frac{1}{2}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-
m_{\psi}\left(1+i\frac{\phi}{f}-\frac{\phi^{2}}{2f^{2}}+...\right)\psi\psi^{c}$
$\displaystyle+\frac{1}{m_{\rho}^{2}f^{2}}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi
m_{\psi}\left(1+i\frac{\phi}{f}-\frac{\phi^{2}}{2f^{2}}+...\right)\psi\psi^{c}+...+\text{h.c.}\,.$
(5.11)

The corrections to $\delta m^{2}_{\phi}$ from the quadratic
$\phi^{2}\psi\psi^{c}$ are exactly canceled by the corrections from the other
interactions. As already mentioned, this cancellation is guaranteed by the
shift symmetry of $\phi$, which is a non-linear realization of the original
$U(1)_{\Phi}$ symmetry. This non-linearly $U(1)_{\Phi}$ symmetry also forbids
any potential of $\phi$ as well which is manifested in a different basis. To
see how it works, one can do an axial field redefinition as

$\psi\longrightarrow e^{i\frac{\phi}{2f}}\psi\,,\qquad\psi^{c}\longrightarrow
e^{i\frac{\phi}{2f}}\psi^{c}\,,$ (5.12)

which yields the following effective Lagrangian

$\mathcal{L}_{\text{EFT}}^{\prime}\supset\mathcal{L}_{\text{SM}}+\frac{1}{2}\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-
m_{\psi}\psi\psi^{c}-\frac{1}{f}\partial_{\mu}\phi
J^{\mu}_{A}+\frac{1}{m_{\rho}^{2}f^{2}}\partial_{\mu}\phi\partial^{\mu}\phi
m_{\psi}\psi\psi^{c}+...+\text{h.c.}\,,$ (5.13)

where
$J^{\mu}_{A}=\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+\bar{\psi^{c}}\bar{\sigma}^{\mu}\partial_{\mu}\psi_{e}^{c}$
is the axial current. Note that we ignored any anomalies as these can be
eliminated by an appropriate choice for the $U(1)_{\Phi}$ charges of the other
fermions. Even in the absence of anomalies, the shift symmetry would be
explicitly broken by a soft mass term for $\phi$, related to its nature as a
DM candidate. Therefore, by using the EOM for $\phi$, one can replace, to
leading order, $\partial_{\mu}\phi\partial^{\mu}\phi\rightarrow
m_{\phi}^{2}\phi^{2}$, yielding an interaction term similar to the one in Eq.
(4.1)

$\displaystyle\mathcal{L}^{\text{nat}}_{\text{int}}=\frac{1}{\Lambda^{2}f^{2}}m_{\psi}\partial_{\mu}\phi\partial^{\mu}\phi\psi\psi^{c}\longrightarrow\frac{m^{2}_{\phi}}{\Lambda^{2}f^{2}}m_{\psi}\phi^{2}\psi\psi^{c}\,.$
(5.14)

As expected, this implies that a natural $d^{(2)}_{m_{i}}$ coupling would be
proportional to $m_{\phi}^{2}$, protecting $m_{\phi}$ against radiative
corrections.
The model presented above is usually discussed in the context of ALPs. As we
are interested in the (pseudo) scalar-electrons coupling, we note that it is
strongly constrained by stellar evolution consideration, as those couplings
provide alternative channels for stellar energy loss processes Raffelt:2012sp
; Redondo:2013lna ; Hardy:2016kme . For instance, the most stringent bound on
the pseudo-scalar electron-Yukawa coupling, $\mathcal{L}\supset
ig^{p}_{e}\phi\bar{\psi_{e}}\gamma^{5}\psi_{e}$, is $g_{e}^{p}\lesssim 3\times
10^{-13}\,$, obtained from the evolution of red giants Raffelt:2012sp . This
can be translated to a bound on the ALP decay constant as $f\gtrsim 2\times
10^{9}\,\,{\rm GeV}\,$. The temporal oscillation of the mass of electron for
an ALP decay constant allowed by cosmological consideration can be written as

$\frac{\delta m_{e}}{m_{e}}\simeq\frac{\rho_{\rm DM}}{f^{4}}\lesssim 2\times
10^{-79}\,\left(\frac{2\times 10^{9}\,{\rm
GeV}}{f}\right)^{4}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)\,,$ (5.15)

which for detection, requires unrealistically high precision from the current
and the near future DDM searches Safronova:2017xyt .

Below, we present different types of light scalar models which do not generate
a linear axion-like interaction of the form $\frac{1}{f}\partial_{\mu}\phi
J^{\mu}_{A}$, or in which such linear interaction is suppressed, and are thus
significantly less constrained. In these models, $\phi$ mainly interacts with
the SM via its quadratic derivatives. These interactions can be naturally
obtained from the kinetic mixing between $\phi$ and its radial mode, as shown
in Eq (5.10).

One example of such realization which leads to quadratic $\phi^{2}$ couplings
without linear couplings is to consider the $\mathbb{Z}_{2}^{\phi}$ even
coupling of a complex scalar field $\Phi$ as

$\mathcal{L}_{\Phi-\rm SM}=\mathcal{L}_{\rm
SM}+\partial^{\mu}\Phi^{\dagger}\partial_{\mu}\Phi+V(\Phi^{\dagger}\Phi)+16\pi^{2}\frac{\Phi^{\dagger}\Phi}{\Lambda}\psi\psi^{c}\,,$
(5.16)

where $\psi,\psi^{c}$ are some SM fermionic fields with mass $m_{\psi}$,
$\Lambda$ is the cut-off scale of the effective interaction. As explained in
Section 4, imposing a $U(1)_{\Phi}$ symmetry that acts solely on $\Phi$
ensures that $\Phi$ may only couple to the SM through powers of
$\Phi^{\dagger}\Phi\,\mathcal{O}^{s}_{\text{SM}}$ or higher derivative powers,
where $\mathcal{O}^{s}_{\text{SM}}$ is an operator contains the SM fields
which is a singlet of all SM symmetries.
Using Eq. (5.10) and the last term of Eq. (5.16), after integrating out the
radial mode $\rho$ at energies below $m_{\rho}$, we get an effective
interaction between $\partial_{\mu}\phi\partial^{\mu}\phi$ and the SM fermion
$\psi$ as,

$\mathcal{L}_{\rm EFT}\supset\frac{16\pi^{2}}{\Lambda
m_{\rho}^{2}}\,\partial_{\mu}\phi\partial^{\mu}\phi\,\psi\psi^{c}\,.$ (5.17)

We require that the effective cutoff would be the largest scale of the EFT,
and thus

$\Lambda\gtrsim{\rm Max}\left[4\pi
f,m_{\psi}\right]\quad\Rightarrow\quad\Lambda\gtrsim{\rm
Max}\left[m_{\rho},m_{\psi}\right]\,.$ (5.18)

Note that the interaction term between the $\Phi$ sector and the SM explicitly
breaks the chiral symmetry and generates a correction to the SM fermion mass
at the tree-level. Assuming no fine cancellation against other contributions
to the fermion mass, we require that the correction to $m_{\psi}$ from Eq.
(5.16) is smaller than the physical value found in experiments. Thus,

$\delta m_{\psi}=\frac{16\pi^{2}}{\Lambda}\frac{f^{2}}{2}\lesssim
m_{\psi}\quad\Rightarrow\quad\Lambda\gtrsim\frac{16\pi^{2}\,f^{2}}{2\,m_{\psi}}\,.$
(5.19)

By the consistency of the Goldstone theory, where $m_{\rho}\lesssim 4\pi f$,
we obtain

$\Lambda\gtrsim\frac{m_{\rho}^{2}}{2\,m_{\psi}}\,\,.$ (5.20)

We note that if the radial mode $\rho$ is lighter than $\psi$, we expect Eq.
(5.18) to give a stronger lower bound on $\Lambda$ than Eq. (5.20) as,

$\Lambda\gtrsim{\rm
Max}\left[\frac{m_{\rho}^{2}}{2\,m_{\psi}},\,\,m_{\rho}\,\,,m_{\psi}\right]\gtrsim
m_{\psi}\,.$ (5.21)

However, as the strength of the $\phi^{2}$ interactions is inversely
proportional to $\Lambda$, raising the cut-off requires a higher experimental
sensitivity to detect the temporal variation of $m_{\psi}$. If we parameterise
the sensitivity of a DDM search experiment in terms of the variation of the
electron mass $(\delta m_{e}/m_{e})$, the maximal $\Lambda$ such an experiment
can probe is

$\Lambda\lesssim 1\,\,{\rm MeV}\left(\frac{10^{-18}}{\left(\delta
m_{e}/m_{e}\right)^{\rm exp}}\right)\left(\frac{30\,\,{\rm
eV}}{m_{\rho}}\right)^{2}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)\,.$ (5.22)

This translate to a bound on the mass of the radial mode, which has to satisfy

$\displaystyle m_{\rho}\lesssim 45\,{\rm eV}\left(\frac{10^{-18}}{\left(\delta
m_{e}/m_{e}\right)^{\rm exp}}\right)^{1/2}\left(\frac{\rho_{\rm
DM}}{\rho_{\text{DM}}^{\odot}}\right)^{1/2}\\!.$ (5.23)

The characteristic sensitivity of $\left(\delta m_{e}/m_{e}\right)^{\rm
exp}\sim 10^{-18}$, would allow probing models with $\rho$ as heavy as
$m_{\rho}\sim 45\,\,{\rm eV}$, saturating the requirement above and a
corresponding maximal cutoff of the same order. We note that an enhanced local
DM density would allow DDM searches to probe models with higher cut-offs and
heavier radial modes accordingly.

###  5.3 The Higgs-portal model - an example of a UV complete theory with no
linear DM couplings

In this section, we provide an example of a specific renormalizable UV model
that could result in the effective low energy SM with additional interactions
of the form of Eq. (1.8). We allow other even orders of derivatives of $\phi$,
such as $(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}$ or
$\square\phi\square\phi$, to be coupled to the SM, but forbid linear
derivative couplings. To achieve the desirable low energy EFT, we extend the
SM field content by introducing a new complex scalar field $\Phi$, which is a
singlet of the SM gauge group. We impose an additional $U(1)_{\Phi}$ global
symmetry, acting only on the $\Phi$ field, under which the SM fields are
neutral.

The most general renormalizable model can be written as

$\displaystyle\mathcal{L}_{\text{UV}}=$
$\displaystyle\>\mathcal{L}_{\text{SM}}+\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi-\lambda_{\Phi
H}\Phi^{\dagger}\Phi H^{\dagger}H-V(\Phi^{\dagger}\Phi)\,,$ (5.24)

where $\mathcal{L}_{\text{SM}}$ denotes the usual SM Lagrangian,
$V(\Phi^{\dagger}\Phi)$ is the potential described in Eq. (5.8) and $H$ is the
SM Higgs doublet. We assume that the potential of $\Phi$ induces a SSB of the
$U(1)_{\Phi}$ symmetry, upon which the low energy description of the theory is
given in terms of the radial and compact modes of $\Phi$, presented in Eq.
(5.9). After the electroweak symmetry breaking, the Higgs portal coupling
induces an interaction between the $\Phi$ sector and the SM fermions through a
mixing of the radial mode $\rho$ with the physical Higgs singlet $h$ as,

$\displaystyle\mathcal{L}_{\text{H-portal}}\simeq\ -\lambda_{\Phi
H}\left(\frac{f+\rho}{\sqrt{2}}\right)^{2}\left(\frac{v_{H}+h}{\sqrt{2}}\right)^{2}\,.$
(5.25)

After diagonalizing the mass matrix $M_{\rho h}$, one obtains a $h-\rho$
mixing angle of

$\displaystyle\sin\theta_{h\rho}\approx\frac{\lambda_{\Phi
H}\,v_{H}\,f}{m^{2}_{h}-m^{2}_{\rho}}\,.$ (5.26)

After integrating out $\rho$ at energy below $m_{\rho}$, we obtain
interactions between $\partial_{\mu}\phi\partial^{\mu}\phi$ and the light SM
fermions as discussed before. In the limit of small mixing, we obtain the
following effective operator

$\displaystyle\mathcal{L}_{\text{EFT}}$
$\displaystyle\simeq\frac{\sin\theta_{h\rho}\,m_{\psi}}{fm^{2}_{\rho}\,v_{H}}\partial_{\mu}\phi\partial^{\mu}\phi\psi\psi^{c}=\frac{\lambda_{\Phi
H}}{\left(m^{2}_{h}-m^{2}_{\rho}\right)}\frac{m_{\psi}}{m^{2}_{\rho}}\partial_{\mu}\phi\partial^{\mu}\phi\psi\psi^{c}\,,$
(5.27)

which is shown diagrammatically in Fig. 14.

$\rightarrow$

Figure 14: A diagrammatic description of the effective operator in the low
energy theory.

While this model is UV complete, it sets a very low upper bound on the mass of
$\rho$. In order for such a model to be detectable considering the current
experimental sensitivity, the mass of the radial mode would have to be
$m_{\rho}\lesssim 10^{-5}\,{\rm eV}$. While such a light invisible scalar that
couples to the Higgs makes this model less appealing, we consider it to be
more of a proof of concept rather than a conclusive case study. In the next
subsection, we would take a more agnostic approach, studying the low-energy
behavior of the natural quadratic coupling without specifying a UV completion.

###  5.4 The clockwork mechanism - an example of a tunable hierarchy between
linear and quadratic couplings

In the previous Section 5.1, we introduced a general EFT approach to linear
and quadratic couplings between the DM $\phi$ and some CP-even SM operators
such as ${\cal O}_{\text{SM}}=F_{\mu\nu}F^{\mu\nu},\;m_{\psi}\psi\psi^{c}\;$
etc. We argued that different operators might break/preserve different
approximate symmetries. Therefore, there could be, in principle, a large
hierarchy between the dimensionless coefficient of different types of
operators. For example, in some models, we expect the linear $\phi$ couplings
to be suppressed and the quadratic couplings of $\phi^{2}$ to the SM to
dominate the physical effects on the induced forces and potential, even though
their naive dimension is higher than the linear couplings.

In this section, we provide two examples where the suppression of the linear
couplings is based on symmetry principles. The examples are based on Clockwork
framework Choi:2015fiu ; Kaplan:2015fuy , and the suppression is due to the
existence of a large hierarchy between the effective periodicity, $F$, compare
to the smaller periodicity $f$, which is the dynamical scale of a spontaneous
symmetry breaking. A detailed description of the Clockwork model can be found
in Appendix D.

In the Clockwork model, the remaining $U(1)_{\text{shift}}$ symmetry, which
keeps the DM mass small, is identified with the remaining
$U\\!\left(1\right)_{\text{clock}}$ of the N+1 Clockwork model, as shown in
Eq. (D.8) where $N$ is the number of Clockwork sites. Note that in the limit
of exact $U(1)_{\text{shift}}$ symmetry, $\phi$ is mass-less. In order to
achieve suppression of the linear $\phi$ couplings, we assign small charges to
the SM fermions, i.e., $[\psi]\sim\mathcal{O}(1)$, such that the leading
symmetry preserving interaction is on the last Clockwork site,

$\mathcal{L}_{\text{int}}^{\text{leading}}\supset
c_{N}\frac{\Phi_{N}}{\Lambda}LHE^{c}+V_{\text{mixing}}\left(\Phi_{N},H\right)+\text{h.c.}\,,$
(5.28)

where $L$ are the left-handed $SU(2)$ doublet SM Lepton fields, while $E^{c}$
are the left-handed $SU(2)$ singlet SM Lepton fields. We assume that
$V_{\text{mixing}}\left(\Phi_{N},H\right)$ breaks CP spontaneously by some
parameter $\theta_{\text{CP}}\ll 1$, which is defined as the CP mixing angle
between the Clockwork and the Higgs CP eigenstates:

$\displaystyle\phi=$
$\displaystyle\cos\theta_{\text{CP}}\hat{\phi}+\sin\theta_{\text{CP}}\hat{h}\,,$
(5.29) $\displaystyle h=$
$\displaystyle-\sin\theta_{\text{CP}}\hat{\phi}+\cos\theta_{\text{CP}}\hat{h}\,.$
(5.30)

$V_{\text{mixing}}\left(\Phi_{N},H\right)$ can give rise to both CP-even and
CP-odd interactions between $\phi$ and the SM fields.

At low energies, we can integrate out the heavy modes of the Clockwork and
obtain an effective Lagrangian of the pNGB which is identified as the ULDM
field,

$\mathcal{L}_{\text{EFT- int}}^{\text{leading}}\supset
c_{N}\frac{f\,e^{i\frac{\phi}{F}}}{\Lambda}h\psi\psi^{c}+V_{\text{mixing}}\left(\frac{\phi}{F},H\right)+\text{h.c.}\,,$
(5.31)

where $c_{N}$ is some $\mathcal{O}(1)$ coefficient. As a result, the
derivative coupling of $\phi$, of the form
$\frac{f}{\Lambda}\frac{\partial_{\mu}\phi}{F}m_{\psi}\left[\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\psi}^{c}\bar{\sigma}^{\mu}\psi^{c}\right]$,
is highly suppressed by $F=3^{N}f$.

The higher dimensional operator

$\mathcal{L}_{\text{int}}^{\text{NL}}\supset
c_{0}\frac{\Phi_{0}^{\dagger}\Phi_{0}}{\Lambda^{2}}m_{\psi}\psi\psi^{c}+\text{h.c.}\,,$
(5.32)

gives rise to a quadratic coupling that is only suppressed by $f$, not $F$.
$c_{0}$ is some $\mathcal{O}(1)$ coefficient. After integrating out the heavy
Clockwork radial modes, from Eq. (5.32) we obtain,

$\mathcal{L}_{\text{EFT- int}}\supset
c_{0}\frac{1}{2f^{2}\Lambda^{2}}\partial_{\mu}\phi\partial^{\mu}\phi\,m_{\psi}\psi\psi^{c}\,.$
(5.33)

Since $F=3^{N}f$, is just an artifact scale of the Clockwork model, it can be
larger than the cutoff scale, $\Lambda\gtrsim f$. Thus, we achieve a hierarchy
between the linear and the quadratic (dimensionless) couplings.

Moreover, adding the both the interactions of Eq. (5.28) and Eq. (5.32) leads
to a collective breaking of the $U\\!\left(1\right)_{\text{clock}}$ symmetry.
As a result, one can write the effective potential generated for this pNGB.
The 1-loop Coleman-Weinberg (CW) effective potential of $\phi$ can be written
as,

$\\!\\!\\!\\!V_{\text{CW}}\left(\phi\right)=\frac{\left(-1\right)^{\mathcal{F}}}{64\pi^{2}}Tr\left[2M^{\dagger}\left(\phi\right)M\left(\phi\right)\Lambda_{c}^{2}+\left(M^{\dagger}\left(\phi\right)M\left(\phi\right)\right)^{2}\ln\frac{M^{\dagger}\left(\phi\right)M\left(\phi\right)}{\tilde{\Lambda}_{c}^{2}}\right]\\!.\\!\\!$
(5.34)

In the equation above, Tr is performed over all field degrees of freedom,
$(-1)^{\mathcal{F}}$ is $+1$ for bosons and $-1$ for fermions. The momentum
cutoff $\Lambda_{c}$, is taken to be of the order of the radial mode mass,
which is of $\mathcal{O}(f)$. Thus, at the leading order the effective
potential of $\phi$ can be approximately written as

$V(\phi)\simeq-
c_{0}c_{N}\frac{m_{\psi}^{2}f^{3}}{16\pi^{2}\Lambda}\cos\left(\frac{\phi}{F}\right).$
(5.35)

The induced DM potential of Eq. (5.35) generates a contribution to the DM mass
of

$m_{\phi}^{2}\simeq
c_{0}c_{N}\frac{m_{\psi}^{2}f^{3}}{16\pi^{2}\Lambda}\frac{1}{F^{2}}\,.$ (5.36)

Therefore, if Eq. (5.35) is the only source of the DM mass, the quadratic
interaction will also be suppressed by $1/F^{2}$, and similar to the linear
interaction. Therefore, we require that there is an additional source of the
DM mass, $m_{\phi}^{2}$ which is not suppressed by $1/F^{2}$.

Another way to employ the Clockwork mechanism is one where the shift symmetry
is broken at the two ends of the Clockwork chain i.e. at the first site, at
$\Phi_{0}$, and at the last site, at $\Phi_{N}$. We assume two different
$Z_{2}$ symmetries are conserved in each site, such that the Lagrangian of
these $U(1)_{\text{clock}}$ breaking sectors can be written as,

$\mathcal{L}_{\text{int}}^{\text{leading}}\supset
c_{N}\frac{\Phi_{N}-\Phi_{N}^{\dagger}}{\Lambda}m_{\psi}\psi\psi^{c}+c_{0}\frac{\Phi_{0}+\Phi_{0}^{\dagger}}{\Lambda}m_{\psi}\psi\psi^{c}\,.$
(5.37)

Therefore, the effective Lagrangian of the pNGB takes the form of

$\mathcal{L}_{\text{effective}}^{\text{leading}}\simeq
c_{N}\frac{f\sin\left(\frac{\phi}{F}\right)}{\Lambda}m_{\psi}\psi\psi^{c}+c_{0}\frac{f\cos\left(\frac{\phi}{f}\right)}{\Lambda}m_{\psi}\psi\psi^{c}\,.$
(5.38)

Assuming the clockwork potential is dominated by a backreaction potential
which has a minimum near $\phi\simeq 0$, one can expand the trigonometric
functions to achieve the desired hierarchy between the linear coupling and the
quadratic coupling, relatively suppressed by $f/F\ll 1$, as

$\mathcal{L}_{\text{effective}}^{\text{leading}}\simeq\frac{c_{N}\,f}{\Lambda
F}\phi m_{\psi}\psi\psi^{c}+\frac{c_{0}}{\Lambda
f}\phi^{2}m_{\psi}\psi\psi^{c}\,.$ (5.39)

##  6 Sensible Models of Light Scalars with Large Quadratic Couplings

As mentioned in previous sections, we are interested in keeping the scalar
naturally light, while also maintaining its quadratic interactions within
experimental reach. This should be achieved in conjunction to ensuring that
the linear scalar interactions are suppressed. In this section we shall
consider two main constructions that realize such a scenario. We begin by
briefly reviewing the idea presented in Hook:2018jle , which involves a QCD
axion with a $\mathbb{Z}_{N}$ symmetry acting on $N$ copies of the SM. We then
move to describe a realization of the relaxed-relaxion idea Banerjee:2020kww ,
that shows that relaxion models may yield naively unnaturally-large couplings
for a light relaxion.

###  6.1 Quadratic interactions of the naturally light $\mathbb{Z}_{N}$ QCD-
axion

It is interesting to note that there is a class of models that naturally leads
to sizeable quadratic interactions between the ULDM and the SM field, with no
corresponding linear scalar coupling. This is precisely the effective theory
of a certain kind of a naturally light QCD-axion model, where the axion mass
is suppressed relative to the conventional models due to the presence of $N$
copies of the SM, which furnish a $\mathbb{Z}_{N}$ symmetry Hook:2018jle ;
DiLuzio:2021pxd  . In this class of models, while the axion couplings to the
SM fields (our SM) follow the conventional models, the axion mass is
suppressed by an additional factor of $z^{N}$, with $z\equiv m_{u}/m_{d}\sim
1/2$ given by the up to down quark mass ratio. Furthermore, as was
demonstrated in Kim:2022ype , while at linear order, the axion couplings to
the SM fields are only to pseudo-scalar operators (thus preserving parity),
the axion also possesses quadratic couplings to the hadrons. This leads to
exciting phenomenology as the QCD axion can be searched for in experiments
that consider the variation of coupling constants instead of the conventional
QCD-axion searches (see e.g. Graham:2013gfa ). For more information, we refer
the reader to Kim:2022ype .

###  6.2 Quadratic interactions of a light relaxed-relaxion

Another possibility to ameliorate the naturalness bound for a light scalar is
by relaxing its mass in a dynamical way Banerjee:2020kww ; Banerjee:2022wzk .
Dynamical relaxation of a light scalar is discussed in the context of the
relaxion mechanism Graham:2015cka , where the light scalar field scans the
Higgs mass parameter starting from some high cut-off down to its measured
value. As discussed Banerjee:2020kww , due to the small incremental change of
the Higgs VEV as a function of the scalar field value, the scalar stops at a
shallow part of its potential and the stopping point (in the field space) is
very close to $\phi/f\sim\pi/2$. As a result of the shallowness of the
scalar’s potential, its mass is suppressed compare to the naive expected
value, however the interaction strength with the Higgs/SM is not. Thus for low
low energy observers, the scalar appears to be unnatural although the relaxion
mechanism is constructed in a technically-natural way.

Below we use a similar idea in order to suppress the mass of the ULDM field,
while maintaining its quadratic interactions observable and suppressing its
linear interactions with the SM. To achieve that we relax the mass of a hidden
sector Higgs, $\hat{H}$, from some cut-off $\Lambda$ to its true VEV
$\hat{v}$, and invoke an interaction of the scalar with the SM as given in Eq.
(6.1).

####  6.2.1 Interaction with the SM

We assume that the field $\phi$ is coupled to the light SM matter as444In
principle there could be some renormalizable portal interaction between the SM
Higgs and $\hat{H}$. Thus through the mixing with the hidden Higgs, $\phi$
will also have some interaction with the SM. However we consider the strength
of such portal interaction to be small and thus this contribution is
negligible.

$\mathcal{L}_{\text{int
}}\supset-\sin\left(\frac{\phi}{f}\right)\left[\sum_{\psi=e,u,d}g_{\phi\psi}m_{\psi}\psi\psi^{c}+\frac{g_{\phi\gamma}}{2}F_{\mu\nu}F^{\mu\nu}+\frac{g_{\phi
g}}{2}G_{\mu\nu}G^{\mu\nu}+\text{h.c.}\right]\,,$ (6.1)

which can be obtained by demanding that the UV completion of this sector
includes a linear complex field
$\Phi=\left(\rho+f\right)\exp\left(i\frac{\phi}{f}\right)$, and respects a
version of charge conjugation under which $\Phi\rightarrow-\Phi^{*}$. At
leading order, Eq. (6.1) generates a quadratic coupling to the light matter in
a technically natural way. We can calculate the 1-loop potential of $\phi$,
generated by the diagrams presented in Fig. 15, similarly to Eq.(5.34), as

$V\left(\phi\right)\simeq\frac{1}{16\pi^{2}}\left[\\!\sum_{\psi=e,u,d}\\!\\!\\!2\,g_{\phi\psi}\,m_{\psi}^{2}\Lambda_{\rm
cut}^{2}\\!\left[\sin\left(\frac{\phi}{f}\right)\\!+\\!g_{\phi\psi}\sin^{2}\left(\frac{\phi}{f}\right)\right]\\!\\!+\\!\\!\sum_{A=\gamma,g}\\!\\!g_{\phi
A}\Lambda_{\rm cut}^{4}\sin\left(\frac{\phi}{f}\right)\\!\right]\\!,$ (6.2)

where $\Lambda_{\rm cut}\gtrsim m_{e}$ is the momentum cut-off of the loop.
The overall potential for $\phi$ can be written as,

$\displaystyle V(\phi,\hat{H})$
$\displaystyle=-g\Lambda^{3}\phi+\left(\Lambda^{2}-g\Lambda\phi-\mu_{\rm
br}^{2}\cos\frac{\phi}{f}\right)|\hat{H}|^{2}+|\hat{H}|^{4}$
$\displaystyle+\frac{1}{8\pi^{2}}\,\sum_{\psi=e,u,d}g_{\phi\psi}m^{2}_{\psi}\Lambda_{\rm
cut}^{2}\,\sin\frac{\phi}{f}+\frac{1}{16\pi^{2}}\sum_{A=\gamma,g}g_{\phi
A}\Lambda_{\rm cut}^{4}\sin\frac{\phi}{f}\,,$ (6.3)

$\hat{H}$ belongs to a new hidden sector whose mass is being relaxed from some
cut-off $\Lambda$ to its true vacuum expectation value
$\hat{v}=\left<\hat{H}\right>$, $g$ is small theory parameter and $\mu_{\rm
br}\lesssim\hat{v}$ is the scale of the cosine potential (for more details of
this kind of construction see e.g. Graham:2015cka ; Banerjee:2020kww ).

\+

Figure 15: Feynman diagrams for generating Coleman Weinberg for $\phi$ at
1-loop order coming from Eq. 6.1.

To achieve a successful relaxation of the mass parameter of $\hat{H}$, we
require

$\mu_{\rm
br}^{2}\hat{v}^{2}\gtrsim\frac{1}{8\pi^{2}}\,g_{\phi\psi}\,m_{\psi}^{2}\,\Lambda_{\rm
cut}^{2}\,.$ (6.4)

Minimizing Eq. (6.3) with respect to both $\hat{H}$ and $\phi$, and using the
above constraint, we find the field stopping point as

$\frac{\phi_{0}}{f}\simeq\frac{\pi}{2}-\frac{\mu_{\rm
br}^{2}}{2\hat{v}^{2}}-\frac{g_{\phi\psi}\,m_{\psi}^{2}\,\Lambda_{\rm
cut}^{2}}{8\pi^{2}\mu_{br}^{2}\hat{v}^{2}}\pm\frac{\mu_{\rm
br}}{\Lambda}\equiv\frac{\pi}{2}-\delta_{\pi/2}\,,$ (6.5)

where $\delta_{\pi/2}\ll 1$ is defined as the deviation from $\pi/2$. As
explained in Banerjee:2020kww , we obtain the mass of the light scalar $\phi$
as,

$\displaystyle m_{\phi}^{2}\approx\frac{\mu_{\rm
br}^{2}\hat{v}^{2}}{f^{2}}\frac{\mu_{\rm br}}{\Lambda}=\frac{\mu_{\rm
br}^{2}\hat{v}^{2}}{f^{2}}\delta\,,$ (6.6)

which is suppressed compare to the naive expected value of $m_{\rm
naive}^{2}\sim\mu_{\rm br}^{2}\hat{v}^{2}/f^{2}$ by a small parameter defined
as $\delta=\mu_{\rm br}/\Lambda\ll 1$. This suppression is a result of the
flatness of the effective potential of $\phi$, which characterizes the first
point at which the derivative of the backreaction potential, mainly controlled
by a periodic function of $\phi$ with a slowly rising amplitude, balances out
the constant derivative of the UV term.

Expanding Eq. (6.1) around the stopping point of $\phi$ gives rise to linear
and quadratic couplings of $\phi$ to the SM as

$\displaystyle\mathcal{L}_{\rm int}\supset$
$\displaystyle-\sum_{\psi=e,u,d}g_{\phi\psi}m_{\psi}\psi\psi^{c}\left(1-\frac{\delta_{\pi/2}^{2}}{2}-\frac{\phi^{2}}{2f^{2}}-\delta_{\pi/2}\frac{\phi}{f}\right)$
$\displaystyle+\left[\frac{g_{\phi\gamma}}{2}F_{\mu\nu}F^{\mu\nu}+\frac{g_{\phi
g}}{2}G_{\mu\nu}G^{\mu\nu}\right]\left(1-\frac{\delta_{\pi/2}^{2}}{2}-\frac{\phi^{2}}{2f^{2}}-\delta_{\pi/2}\frac{\phi}{f}\right)\,.$
(6.7)

Moreover, for a valid weakly coupled theory, we require

$\displaystyle\delta m_{\psi}\lesssim m_{\psi}\qquad\text{
i.e}\>g_{\phi\psi}\lesssim\mathcal{O}(1)\,,$ (6.8)
$\displaystyle\delta\alpha_{(s)}\lesssim\alpha_{(s)}\qquad\\!\\!\text{i.e.
}\>g_{\phi\gamma(g)}\lesssim\mathcal{O}(1)\,.$ (6.9)

Combining Eq. (6.4) and Eq. (6.9) we get

$\displaystyle g_{\phi\psi}\lesssim{\rm Min}\left[1,8\pi^{2}\frac{\mu_{\rm
br}^{2}\hat{v}^{2}}{m_{\psi}^{2}\,\Lambda^{2}}\right]\,,$ (6.10)
$\displaystyle g_{\phi\gamma(g)}\lesssim{\rm
Min}\left[1,4\pi^{2}\frac{\mu_{\rm br}^{2}\hat{v}^{2}}{\Lambda^{4}}\right]\,.$
(6.11)

####  6.2.2 Relaxation of the cutoff

Consider a model where the field stopping point is not near
$\phi_{0}/f\sim\frac{\pi}{2}$ as in Eq. (6.5), but a naive order one stopping
point,

$\left(\frac{\phi_{0}}{f}\right)_{\text{Naive}}\sim\mathcal{O}(1)\,.$ (6.12)

For simplicity, let us focus on the quadratic coupling of $\phi$ to the
electron

$\mathcal{L}\supset g_{\phi e}\frac{\phi^{2}}{f^{2}}\,m_{e}\,ee^{c}\,.$ (6.13)

Given the naive stopping point, this interaction leads to a quadratic
contribution to the scalar mass as

$\left(\delta m_{\phi}^{2}\right)_{\text{Naive}}\simeq g_{\phi
e}\,m_{e}^{2}\,\frac{\Lambda_{e}^{2}}{8\pi^{2}f^{2}}\,,$ (6.14)

where $\Lambda_{e}$ is the cut-off of the electron loop. In addition, if the
scalar field $\phi$ accounts for the DM in the present universe, it acquires a
time-dependent background value. Thus, the same coupling would induce temporal
variations of the mass of the electron as

$\frac{\delta m_{e}}{m_{e}}(t)\simeq g_{\phi e}\frac{\rho_{\rm
DM}}{f^{2}m_{\phi}^{2}}\,.$ (6.15)

We then obtain a simple relation between the variation of the fundamental
constant and the correction to the mass of $\phi$

$\left(\delta m_{\phi}^{2}\right)_{\text{Naive}}=\left(\frac{\delta
m_{e}}{m_{e}}\right)(t)\times\frac{m_{e}^{2}\,m_{\phi}^{2}\,\Lambda_{e}^{2}}{\rho_{\rm
DM}\,8\pi^{2}}\,.$ (6.16)

Without the relaxed relaxion dynamics, a theory involving such interactions
can only be natural if

$\left(\delta m_{\phi}^{2}\right)_{\text{Naive}}\lesssim
m_{\phi}^{2}\Rightarrow\frac{\delta
m_{e}}{m_{e}}(t)\times\frac{\Lambda_{e}^{2}}{8\pi^{2}}\lesssim\frac{\rho_{\rm
DM}}{m_{e}^{2}}\,.\\\ $ (6.17)

Thus, naively, theories of natural light scalars generating observable
variations of constants of nature would imply new physics should appear at
relatively low scales as

$\displaystyle\Lambda_{e}\lesssim 30\,{\rm eV}\,\left(\frac{\rho_{\rm
DM}}{\rho_{\rm DM}^{\odot}}\right)^{1/2}\left(\frac{10^{-18}}{\left(\delta
m_{e}/m_{e}\right)^{\rm exp}}\right)^{1/2}.$ (6.18)

In light of this consideration, we can determine whether a theory described by
Eq. (6.3) is natural or unnatural, by quantifying by how much the cut-off of
the theory is being relaxed away from the naive estimation above.

In the relaxed scenario, the 1-loop order corrections to the mass of $\phi$
are already taken into account via the effective potential in Eq. (6.3),
yielding the stopping point given in Eq. (6.5), and the suppressed mass in Eq.
(6.6). Therefore, naturalness does not require that these 1-loop corrections
are kept smaller than $m_{\phi}$. However, for the relaxation to be successful
(both of the mass of $\hat{H}$ and of the mass of $\phi$), we must require
that the coupling of $\phi$ to the SM does not overcome the derivative of the
$\hat{H}$-dependent potential of $\phi$ at the first stopping point. This
requirement is expressed in Eq. (6.4), and can be re-expressed as

$\displaystyle\frac{{g}_{\phi
e}\,m_{e}^{2}\,\Lambda_{e}^{2}}{8\pi^{2}f^{2}}\lesssim\frac{\mu_{\rm
br}^{2}\hat{v}^{2}}{f^{2}}\,,$ (6.19)

or

$\displaystyle\left(\delta
m_{\phi}^{2}\right)_{\rm{Naive}}\lesssim\frac{m_{\phi}^{2}}{\delta}\,,$ (6.20)

which is less restrictive than the naive requirement in Eq. (6.17) since
$\delta\lesssim 1$. Consequently, the phenomenological effects associated with
the coupling of $\phi$ to the SM may be enhanced, without significantly
altering the mass of the scalar. Namely, the cut-off scale of such a theory
$\Lambda_{e}$, including a light scalar and observable quadratic interactions
with the SM, is relaxed by a factor of $1/\sqrt{\delta}$ with respect to the
naive prediction of Eq. (6.18).

####  6.2.3 A large hierarchy between the linear and quadratic coupling due to
the Relaxed Relaxion mechanism

A large hierarchy between the effective linear and quadratic couplings of
$\phi$ can be simply parameterize in terms of the deviation of the scalar’s
stopping point from $\frac{\pi}{2}$, denoted by $\delta_{\pi/2}$. The
different scaling between the dimensionless linear and quadratic couplings is

$c_{\text{linear}}\sim
g_{\phi(\psi,A)}\times\delta_{\pi/2},\qquad\text{while}\qquad
c_{\text{quadratic}}\sim g_{\phi(\psi,A)}\,.$ (6.21)

In our natural theory we consider
$g_{\phi(\psi,A)}\sim\mathcal{O}\left(1\right)$. Moreover, we wish to have
natural values of the scaled couplings, which gives new bounds that are absent
if only either the linear or the quadratic coupling were to be turned on. For
example, in order to match the observed bound of
$d_{m_{e}}^{\left(2\right)}\sim 10^{14}$, one must require that

$\mathcal{O}(1)\sim
c_{m_{e}}^{\left(2\right)}=d_{m_{e}}^{\left(2\right)}\times\frac{f^{2}}{M_{\text{pl}}^{2}}\simeq\frac{f^{2}}{\text{GeV}^{2}}\times
10^{-23}\;\Rightarrow\;f\simeq 3\times 10^{11}\,\text{GeV}\,,$ (6.22)

in order to get $\mathcal{O}(1)$ coupling. This, in turn, sets the value of
the relaxion dynamical scale, $f$. For the linear coupling, matching the
observed bound on $d_{m_{e}}^{\left(1\right)}\sim 10^{-3}$ translates to

$c_{m_{e}}^{\left(1\right)}=d_{m_{e}}^{\left(1\right)}\times\frac{f}{M_{\text{pl}}}\simeq
10^{-10}\sim\delta_{\pi/2}$ (6.23)

which sets the value of $\delta_{\pi/2}$. Assuming no cancellation between the
different terms contributing to $\delta_{\pi/2}$, we get the following
constraints:

$\begin{cases}\frac{\mu_{{\rm br}}^{2}}{\hat{v}^{2}}\lesssim 10^{-10}\,\\\
\frac{\mu_{{\rm br}}}{\Lambda}\lesssim 10^{-10}\,\\\
\frac{g_{\phi\psi}\,m_{\psi}^{2}\,\Lambda_{\rm
cut}^{2}}{8\pi^{2}\mu_{br}^{2}\hat{v}^{2}}\lesssim 10^{-10}\,.\end{cases}$
(6.24)

The first two constraints from Eq. (6.24) can be easily satisfied by
constructing $\mu_{{\rm br}}\ll\hat{v},\Lambda$. The last constraint from Eq.
(6.24) can also be achieved. However, it suggests that it is more likely to
see an effect of both linear and quadratic couplings to light SM matter.

An example of the allowed parameter space for this model, including the
theoretical considerations above as well as the experimental bounds, is
presented in Fig 16 for a fixed $f=10\,\hat{v}=10^{7}\,\text{GeV}$ and a
cutoff of $\Lambda_{e}=10\,\text{MeV}$. We plot both the bounds on the
quadratic coupling and on the linear coupling (for a generic
$\mathcal{O}\left(1\right)$ stopping point and for the relaxed-relaxion
mechanism) separately, however, since they are not independent in this model,
the bound should be drawn from satisfying these constraints simultaneously
(shaded lilac region of Fig. 16). This yields non-trivial constraints both due
to the interdependence of these couplings in this model, making it impossible
to turn off the quadratic coupling without turning off the linear coupling as
well, and also taking into account the non-linear effects discussed in
subsection 3.1. Note that the EP constraints involving the quadratic coupling
strongly depend on on the radius of the source $R_{C}$ and on the distance
from the center of the source $r$. Our bounds are calculated assuming
$R_{C}=r=R_{\oplus}$ for Eöt-Wash Be/Ti Schlamminger:2007ht  and
$R_{C}=R_{\oplus}\,,r=R_{\oplus}+700$ km for MICROSCOPE PhysRevLett.129.121102
, with $R_{\oplus}$ being the radius of the Earth. Although we have focused on
the electron coupling $g_{\phi e}$, the same analysis can be easily extended
to other couplings between $\phi$ and the SM fields, such as
$g_{\phi\gamma},g_{\phi u(d)}$ and $g_{\phi g}$.

Figure 16: The allowed parameter space of an ULDM relaxed-relaxion inspired
model with $f=\Lambda=10\,\hat{v}=10^{7}\text{\,GeV}$, and
$\Lambda_{e}=10\,\,{\rm MeV}$. Blue - bounds on a model with only the
suppressed linear coupling given, purple - bounds on a model with only the
quadratic coupling, light blue - bounds on a model with a naive linear
coupling, lilac - bounds given by the combination of the quadratic coupling
and the suppressed linear coupling. Pink - the critical coupling for the
quadratic interactions. Black - the region in which there is no relaxation of
the mass parameter of $\hat{H}$, gray - region in which the ULDM amplitude is
greater than $f$.

##  7 Conclusions

In this work, we considered a special class of spin-0 ULDM models. In this
class, the dominant operators describing the interactions between the ULDM
field $\phi$ and the SM are quadratic in the ULDM field, namely, of the form
$\phi^{2}\mathcal{O}_{\rm SM}$ where $\mathcal{O}_{\rm SM}$ composed of SM