aainstitutetext: Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germanybbinstitutetext:  Institute for Particle Physics
Phenomenology, Durham University, Durham DH1 3LE, UKccinstitutetext:
Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Herzl St 234, Rehovot 761001, Israel

# Lorentz violating backgrounds from quadratic, shift-symmetric, ultralight
dark matter

Minyuan Jiang b Benjamin D. Pecjak c Gilad Perez c Somasundaram
Sankaranarayanan

###### Abstract

We consider an effective theory for a shift-symmetric, quadratically-coupled,
ultralight spin-0 field. The leading CP conserving interactions with Standard
Model fields in the effective theory arise at dimension 8. We discuss the
renormalization group evolution and positivity bounds on these operators, as
well as their possible UV origins. Assuming that the spin-0 field is
associated with an ultralight dark matter candidate, we discuss the effects of
the dimension-8 operators on experiments searching for the oscillation of
fundamental constants and Lorentz violation. We find that the direct bounds on
these two effects are of similar strength but rather weak, corresponding to a
UV cutoff scale of keV order, as they are mediated by dimension-8 operators.

††preprint: IPPP/24/21, DESY-24-056

##  1 Introduction

The possibility that dark matter (DM) consists of ultralight bosonic particles
with sub-eV mass has garnered increasing interest in recent times. The small
mass of these particles results in a large occupation number, so that rather
than behaving as discrete particles, these particles exhibit collective
behavior best described by classical waves. A spin-0 ultralight field can
generally lead to a viable model of DM via the misalignment mechanism
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah . The field $\phi$ can be CP-
odd, such as the QCD axion Peccei:1977hh ; Weinberg:1977ma ; Wilczek:1977pj ;
Kim:1979if ; Shifman:1979if ; Dine:1981rt ; Zhitnitsky:1980tq  that was
originally proposed to address the strong CP problem, or axion-like particles
predicted in other well motivated models of spontaneous U(1) symmetry breaking
(see for instance Froggatt:1978nt ; Gelmini:1980re ). Alternatively, it could
be CP-even, as exemplified by dilatons arising from the spontaneous breaking
of a conformal sector, or otherwise a pseudo/scalar, ultralight DM (ULDM),
hybrid with ill-defined parity transformation, which arises in axion models
with more than one source of shift symmetry breaking Flacke:2016szy ;
Banerjee:2018xmn .

In any case, the specific UV origins of these particles are not our primary
concern, as in the present paper we leverage the framework of Effective Field
Theory (EFT) to describe their interactions at low energies systematically.
Our EFT framework is characterized by two symmetries, which require the
invariance of the theory under (1) the shift
$\phi\rightarrow\phi+\textbf{const.}$111Note that this symmetry is softly
broken by the mass term. and (2) a $\mathbb{Z}_{2}$ transformation
$\phi\rightarrow-\phi$. The shift symmetry protects the DM mass from large
corrections and keeps it naturally small, and the $\mathbb{Z}_{2}$ symmetry
implies that $\phi$ couples quadratically to the Standard Model (SM) fields.
In addition, we consider only CP-even operators. In fact, in models where the
SM fields are not charged under the U(1) axionic symmetry, we expect
generically that the leading interactions between the ULDM and the SM would
appear at quadratic order, without the need to impose a $\mathbb{Z}_{2}$
symmetry, and on top of that one should add that almost all models of an
axion-like particle are expected to consist of quadratic coupling to the SM
scalar operators Banerjee:2022sqg .

Our EFT provides a consistent framework for describing Lorentz and in
particular Galilean violating backgrounds, which we shall refer to
collectively as Lorentz violating (LV). We shall demonstrate that it is
comprised of two distinct types of effective operators governing the
interactions between the field $\phi$ and SM fields. One type possesses a
scalar structure, leading to the oscillation of fundamental constants such as
the electron mass and the fine structure constant, while the other exhibits a
tensor structure, resulting in similar signals in experiments searching for
the aforementioned effects, and in addition the violation of the Lorentz
symmetry (in particular the violation of spatial rotation symmetry).
Consequently, the EFT links two classes of experiments probing for these
different effects, which are usually considered to be unrelated. Such a
theoretical link is timely, as experiments searching for oscillations of
fundamental constants by comparing the frequencies in atomic clocks and
cavities have undergone significant developments in recent years (see
Antypas:2022asj  for a review of these experiments), as have experiments
focusing on searches for LV effects predicted in theories of quantum gravity
(see Kostelecky:2008ts  for a summary of the experiments and results). Since
the experiments we are concerned with (in particular the searches for LV)
focus on the Quantum Electrodynamics (QED) sector of the SM, in this work we
will focus on the operators built from the $\phi$, electron and photon fields.

This paper is organized as follows. In section 2, we study the theoretical
aspects of the EFT. We first list all the relevant operators and then study
the RG flow between them. Additionally, we give an example of potential UV
origin of these operators, and explore the constraints imposed on the Wilson
coefficients by positivity bounds. In section 3, we demonstrate how these
effective operators affect the aforementioned two types of experiment and
derive bounds on the Wilson coefficients. A brief discussion of indirect
bounds from astrophysical, cosmological and collider data is presented in
section 4, and conclusions are given in section 5.

##  2 EFT of ULDM with quadratic coupling to QED

In this section we study the EFT that describes the interaction between a real
spin-0 field $\phi$ and the QED sector of the SM. The scalar is assumed to be
a singlet under the SM gauge group. As already described, we also assume that
the theory is invariant under CP, shift symmetry, and a $\mathbb{Z}_{2}$,
$\phi\rightarrow-\phi$.

###  2.1 Effective operators

The EFT is organized as an expansion in $1/\Lambda$, with $\Lambda$
characterizing the UV cut-off of the theory. The effective operators of higher
mass dimension are thus suppressed by higher orders of $1/\Lambda$. In our
case, the first operator that satisfies all symmetries of the theory arises at
dimension seven, and is given by

$\overline{O}_{1}=\partial_{\mu}\phi\partial^{\mu}\phi\,\bar{\psi}\psi\,,$ (1)

where $\psi$ stands for a fermion field (for instance, an electron).

At dimension 8 there are four independent CP-even operators 222Other than
these CP-even operators, there are also CP-odd ones that obey the
$\mathbb{Z}_{2}$ and shift symmetries but we do not consider them here., in
the sense that they are not related to each other by the equations of motion
(EoM) or integration by parts. These are

$\displaystyle O_{1}$
$\displaystyle=\partial_{\mu}\phi\partial^{\mu}\phi\,i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\nu}}\psi,$
$\displaystyle O_{2}$
$\displaystyle=\partial_{\mu}\phi\partial_{\nu}\phi\,i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D^{\mu}}\psi,$
$\displaystyle O_{3}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi
F_{\nu\rho}F^{\nu\rho}\,,$ $\displaystyle O_{4}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi{F^{\mu}}_{\rho}F^{\nu\rho}\,.$
(2)

The notation is such that
$\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D^{\mu}}\psi\equiv\frac{1}{2}\bar{\psi}\gamma^{\nu}D^{\mu}\psi-\frac{1}{2}\overline{(D^{\mu}\psi)}\gamma^{\nu}\psi$,
$D_{\mu}\equiv\partial_{\mu}-ieA_{\mu}$ with $A_{\mu}$ the vector potential of
the photon and $e$ the charge of electron, and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the gauge
invariant EM field tensor.

The operators $O_{1}$ and $\overline{O}_{1}$ can be transformed into each
other using the EoM $i\gamma^{\mu}D_{\mu}\psi=m_{\psi}\psi$. This means that
by a redefinition of the field $\psi$, one can eliminate $\overline{O}_{1}$,
at the expense of shifting the Wilson coefficient of $O_{1}$ (see Appendix A
for more details). Due to the soft breaking of shift symmetry from the mass
term, there could also be operators of the form
$\phi^{2}i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\nu}}\psi$ and
$\phi^{2}F_{\nu\rho}F^{\nu\rho}$, suppressed by $m_{\phi}^{2}/\Lambda^{2}$.
When taking $\phi$ as the DM background, these two operators have the same
effects as $O_{1}$ and $O_{3}$ for the experiments we consider. They will not
mix with the dimension-8 operators under RG evolution. We therefore ignore
these two operators in most of the following discussion, but point out when
they are relevant.

Thus our effective Lagrangian is given by

$\mathcal{L}_{\text{eff}}=i\bar{\psi}(\not{D}-m_{\psi})\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\sum_{i=1}^{4}c_{i}O_{i}\,.$
(3)

Notice that with this definition the Wilson coefficients $c_{i}$ carry mass
dimension minus four. When interpreting the bounds on the Wilson coefficients
from experimental data discussed later on, it will sometimes be useful to make
the mass dependence of the Wilson coefficients explicit by writing

$\displaystyle c_{i}=\frac{16\pi^{2}}{\Lambda^{4}}\,,$ (4)

where $\Lambda$ is a mass scale and the factor of $16\pi^{2}$ follows the
conventions of naive dimensional analysis (see e.g.Jenkins:2013sda ). Using
this equation, a given bound on a Wilson coefficient can be translated into a
mass scale being indirectly probed by the measurement.

###  2.2 Renormalization-group flow

The Wilson coefficients $c_{i}\equiv c_{i}(\mu)$ are functions of the
renormalization scale $\mu$. From the top-down point of view, they are
obtained by matching the EFT and the UV theory at the scale $\Lambda$. Then to
use them for processes at a typical low-energy scale $\mu_{\rm exp}$ defined
by experiments, one needs to evolve these coefficients from $\mu=\Lambda$ down
to $\mu=\mu_{\rm exp}$. This evolution is governed by the Renormalization
Group Equations (RGE).

The computation of the RG flow is outlined in Appendix B. For the RGE at one-
loop level we find

$\displaystyle\frac{\rm d}{\rm d\log\mu}\begin{bmatrix}c_{1}\\\ c_{2}\\\
c_{3}\\\
c_{4}\end{bmatrix}=\frac{\alpha}{4\pi}\begin{bmatrix}0&\frac{-4}{3}&12&\frac{7}{3}\\\
0&\frac{16}{3}&0&\frac{8}{3}\\\ 0&\frac{-4}{3}&\frac{8}{3}&0\\\
0&\frac{16}{3}&0&\frac{8}{3}\\\ \end{bmatrix}\begin{bmatrix}c_{1}\\\ c_{2}\\\
c_{3}\\\ c_{4}\end{bmatrix}\,,$ (5)

where $\alpha=e^{2}/(4\pi)$ is the fine structure constant. The non-zero off-
diagonal terms in the above matrix imply that the operators generally mix with
each other under RG evolution. However, we notice that non-trivial patterns
exist in our case and are governed by symmetries. This can be seen explicitly
by changing the operator basis and organizing the operators into two sectors,
characterized by the quantum number $j$ of the angular momentum of the
$(\phi,\phi)$ channel Jiang:2020rwz :

  * •

$j=0$ sector:

$\displaystyle\mathcal{O}_{1}$ $\displaystyle\equiv
O_{1}=\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\nu}}\psi\,,$
$\displaystyle\mathcal{O}_{3}$ $\displaystyle\equiv
O_{3}=\left(\partial_{\mu}\phi\partial^{\mu}\phi\right)\frac{1}{2}F_{\nu\rho}F^{\nu\rho}\,.$
(6)

  * •

$j=2$ traceless sector:

$\displaystyle\mathcal{O}_{2}$ $\displaystyle\equiv
O_{2}-\frac{1}{4}O_{1}=\left(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}\eta_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi\right)i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D^{\mu}}\psi\,,$
$\displaystyle\mathcal{O}_{4}$ $\displaystyle\equiv
O_{4}-\frac{1}{4}O_{3}=\left(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{4}\eta_{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi\right)\frac{1}{2}{F^{\mu}}_{\rho}F^{\nu\rho}\,.$
(7)

The $j=0$ sector is easy to identify since its operators are products of two
scalar operators in the dark and QED sectors, respectively. The spin-2 sector
can be checked by noticing that the contribution from
$\mathcal{O}_{2}$($\mathcal{O}_{4}$) to the amplitude for
$\phi+\phi\rightarrow\psi+\psi$ ($\phi+\phi\rightarrow\gamma+\gamma$)
scattering consists only of partial waves of angular momentum $j=2$. We note
that in the $j=2$ operators, the two $\phi$ fields construct a traceless
tensor.

Written in the new basis, the mixing matrix is (with $\mathcal{C}_{i}$ being
the coefficient of $\mathcal{O}_{i}$)

$\displaystyle\frac{\rm d}{\rm d\log\mu}\begin{bmatrix}\mathcal{C}_{1}\\\
\mathcal{C}_{3}\\\ \mathcal{C}_{2}\\\
\mathcal{C}_{4}\end{bmatrix}=\frac{\alpha}{4\pi}\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}12}&0&0\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{8}{3}}&0&0\\\
0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{16}{3}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{8}{3}}\\\
0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{16}{3}}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{8}{3}}\\\
\end{bmatrix}\begin{bmatrix}\mathcal{C}_{1}\\\ \mathcal{C}_{3}\\\
\mathcal{C}_{2}\\\ \mathcal{C}_{4}\end{bmatrix}\,.$ (8)

The mixing matrix is now block diagonal (colour coded in Eq. (8) with blue and
red), indicating that RG flow between the two sectors is forbidden by angular
momentum conservation. We will work with the new basis defined by
$\mathcal{O}_{i}$ in the following.

There are two additional noteworthy features of the above mixing matrix.
First, in the spin-0 sector (colour coded with blue), the operator
$\mathcal{O}_{1}$ does not generate $\mathcal{O}_{3}$, whereas
$\mathcal{O}_{3}$ does generate $\mathcal{O}_{1}$. This is because
$\mathcal{O}_{1}$ is effectively dimension 7 (see Appendix A) whereas
$\mathcal{O}_{3}$ is dimension 8, and in QFT there is no flow from an operator
of lower dimension to an operator of higher dimension. Second, in the $j=2$
sector, the 2$\times$2 mixing matrix (colour coded with red) has rank 1,
suggesting that there is a linear combination of operators, namely
$\mathcal{O}_{2}-2\mathcal{O}_{4}$, that does not run. Whether this holds only
at one loop level or is also due to some hidden symmetry remains to be
verified.

In section 3 we will derive bounds on the Wilson coefficients from terrestrial
experiments that are sensitive to scales at or below the electron mass. The
Wilson coefficients $\mathcal{C}_{i}(m_{e})$ entering such bounds are related
to those at the scale $\Lambda$ characteristic of a UV completion through the
solution to the RG equation. Neglecting the running of $\alpha$ this solution
is simply

$\displaystyle\mathcal{C}_{i}(m_{e})$
$\displaystyle=\left[\exp\left(\frac{\alpha}{4\pi}\gamma\ln\frac{m_{e}}{\Lambda}\right)\right]_{ij}\mathcal{C}_{j}(\Lambda)\,,$
(9)

where $\gamma$ is the matrix on the right-hand side of Eq. (8). Taking
$\alpha=1/137$ and $\Lambda=1$ TeV as an example, the Wilson coefficients at
the two scales are related by

$\displaystyle\begin{bmatrix}\mathcal{C}_{1}(m_{e})\\\
\mathcal{C}_{3}(m_{e})\\\ \mathcal{C}_{2}(m_{e})\\\
\mathcal{C}_{4}(m_{e})\end{bmatrix}=\begin{bmatrix}\mathcal{C}_{1}(\Lambda)-0.10\mathcal{C}_{3}(\Lambda)\\\
0.98\mathcal{C}_{3}(\Lambda)\\\
0.96\mathcal{C}_{2}(\Lambda)-0.02\mathcal{C}_{4}(\Lambda)\\\
0.98\mathcal{C}_{4}(\Lambda)-0.04\mathcal{C}_{2}(\Lambda)\end{bmatrix}\,.$
(10)

Even for the modest value $\Lambda=1$ TeV, the effects of RG running are as
large as 5–10%, depending on the coefficient.

###  2.3 An example UV completion

The form of the operators, in particular in the $j=2$ sector, indicates that
they can originate from integrating out heavy spin-2 particles from a UV
theory. Massive spin-2 particles generally arise as composite states
(analogous to spin-2 mesons in QCD), or in models with extra dimensions such
as the Kaluza-Klein (KK) graviton modes Kaluza:1921tu ; Klein:1926fj . Well
known examples are the ADD Arkani-Hamed:1998jmv  model and the RS model
Randall:1999ee , motivated as solutions to the hierarchy between the Planck
scale and the weak scale. In the ADD setup, the extra dimensions are flat and
the fundamental scale is on the order of a TeV. The weakness of gravity is due
its leakage into the extra dimensions and to get an effective Planck scale in
4D, the size of extra dimensions are relatively large. So in the ADD case the
KK gravitons are light and the mass gap is small, and show up as collective
effects. In the RS model, the extra dimension is exponentially warped by the
bulk cosmological constant while the 4D branes at the two boundaries remain
flat due to the balance between the cosmological constant and the brane
tension. With the fundamental scale of the theory set to be Planck scale,
fields living on the IR brane (the one at the more curved end of the extra
dimension) will appear to have effective mass of the weak scale. Contrary to
the ADD case, the KK graviton modes in RS have order TeV mass gaps and could
show up as distinct resonances at colliders. Below we are not necessarily
committed to any of these specific scenarios, rather we show that they can
generate the dimension-8 operators contained in our low-energy EFT and thus
provide examples of UV completions.

Figure 1: At energies much lower than the mass of the KK graviton, tree-level
exchange is approximated by a local interaction and represented as an
effective operator.

We thus consider the tree-level exchange of a massive graviton (e.g. the
lightest KK modes in the RS model) between the light scalar $\phi$ and
electrons/photons. At energies much smaller than the mass of the KK graviton,
this interaction can be approximated by local effective operators, see Fig. 1
as an illustration.

This tree-level matching process can be formally carried out by replacing the
heavy field in the action by its classical solution of the EoM. Regardless of
its UV origin, the kinetic part of the action of a massive spin-2 field
$h^{\mu\nu}$ is described by the Fierz-Pauli action Fierz:1939ix ,

$\displaystyle S^{\rm FP}=\int
d^{4}x\left\\{\frac{1}{2}\partial_{\rho}h_{\mu\nu}\partial^{\rho}h^{\mu\nu}\right.$
$\displaystyle-\partial_{\mu}h_{\nu\rho}\partial^{\nu}h^{\mu\rho}+\partial_{\mu}h^{\mu\nu}\partial_{\nu}h$
(11)
$\displaystyle\left.-\frac{1}{2}\partial_{\rho}h\partial^{\rho}h-\frac{1}{2}m_{KK}^{2}\left(h_{\mu\nu}h^{\mu\nu}-h^{2}\right)\right\\}\,,$

where $h=h^{\mu}_{\mu}$. The KK graviton couples to matter fields through the
energy-momentum tensor $T^{\mu\nu}$, with the interaction

$\mathcal{L}_{\text{int}}=-\frac{1}{\Lambda_{KK}}h_{\mu\nu}T^{\mu\nu}\,.$ (12)

Here the scale $\Lambda_{KK}$ characterizes the strength of the interaction.
In the RS model we have $\Lambda_{KK}\sim$ TeV but more generally it is
associated with the scale of spontaneous CFT breaking or the dynamical scale
of confinement in composite models. For fermions ($\psi$), vectors ($\gamma$),
and scalars ($\phi$), the energy-momentum tensor takes the form

$\displaystyle T_{\mu\nu}^{\psi}$
$\displaystyle=\frac{i}{4}\bar{\psi}\left(\gamma_{\mu}D_{\nu}+\gamma_{\nu}D_{\mu}\right)\psi-\frac{i}{4}\left(D_{\mu}\bar{\psi}\gamma_{\nu}+D_{\nu}\bar{\psi}\gamma_{\mu}\right)\psi$
$\displaystyle-\eta_{\mu\nu}\left(\bar{\psi}\gamma^{\rho}D_{\rho}\psi-
m_{\psi}\bar{\psi}\psi\right)+\frac{i}{2}\eta_{\mu\nu}\partial^{\rho}\left(\bar{\psi}\gamma_{\rho}\psi\right)\,,$
(13) $\displaystyle T_{\mu\nu}^{\gamma}$
$\displaystyle=\frac{1}{4}\eta_{\mu\nu}F^{\lambda\rho}F_{\lambda\rho}-F_{\mu\lambda}F_{\nu}^{\,\lambda}\,,$
(14) $\displaystyle T_{\mu\nu}^{\phi}$
$\displaystyle=\partial^{\mu}\phi\partial^{\nu}\phi-\frac{1}{2}\eta^{\mu\nu}\left(\partial_{\rho}\phi\partial^{\rho}\phi-
m_{\phi}^{2}\phi^{2}\right)\,.$ (15)

To integrate out the KK graviton at tree level, we rewrite the full action as

$S=\int
d^{4}x\left\\{\frac{1}{2}h_{\mu\nu}\mathcal{O}^{\mu\nu\rho\sigma}h_{\rho\sigma}-\frac{1}{\Lambda_{KK}}h_{\mu\nu}T^{\mu\nu}\right\\}\,,$
(16)

with

$\mathcal{O}_{\rho\sigma}^{\mu\nu}=-\left[\eta_{(\rho}^{(\mu}\eta_{\sigma)}^{\nu)}-\eta^{\mu\nu}\eta_{\rho\sigma}\right]\left(\square+m_{KK}^{2}\right)+2\partial^{(\mu}\partial_{(\rho}\eta_{\sigma)}^{\nu)}-\partial^{\mu}\partial^{\nu}\eta_{\rho\sigma}-\partial_{\rho}\partial_{\sigma}\eta^{\mu\nu}\,.$
(17)

Then the classical equation of motion for $h_{\mu\nu}$ reads

$\mathcal{O}^{\mu\nu\rho\sigma}h_{\rho\sigma}=\frac{1}{\Lambda_{KK}}T^{\mu\nu}\,,$
(18)

and we obtain the classical solution

$h^{c}_{\rho\sigma}=\mathcal{O}^{-1}_{\rho\sigma\alpha\beta}\frac{1}{\Lambda_{KK}}T^{\alpha\beta}\,,$
(19)

with $\mathcal{O}^{-1}$ defined by

$\mathcal{O}^{\mu\nu\rho\sigma}\mathcal{O}^{-1}_{\rho\sigma\alpha\beta}=\frac{1}{2}(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\beta}\delta^{\nu}_{\alpha})\,.$
(20)

The effective Lagrangian is then obtained by substituting the classical
solution back into the original Lagrangian,

$\displaystyle\mathcal{L}^{\text{EFT}}$
$\displaystyle=-\frac{1}{2\Lambda_{KK}}h^{c}_{\mu\nu}T^{\mu\nu}$
$\displaystyle=-\frac{1}{2\Lambda_{KK}^{2}}T^{\mu\nu}\mathcal{O}^{-1}_{\mu\nu\rho\sigma}T^{\rho\sigma}\,.$
(21)

The effective operators are obtained by expanding $\mathcal{O}^{-1}$ in
$1/m_{KK}^{2}$. To leading order, we have

$\displaystyle\mathcal{O}^{-1}_{\mu\nu\rho\sigma}=-\frac{1}{m_{KK}^{2}}\left(\frac{1}{2}\eta_{\mu\rho}\eta_{\nu\sigma}+\frac{1}{2}\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{1}{3}\eta_{\mu\nu}\eta_{\rho\sigma}\right)\,.$
(22)

Substituting the expressions for $T^{\mu\nu}$ in Eq. (15) into Eq. (21), we
obtain the effective operators

$\displaystyle\mathcal{L}^{\text{EFT}}$
$\displaystyle\supset\frac{1}{2\Lambda_{KK}^{2}m_{KK}^{2}}[(\partial^{\mu}\phi\partial^{\nu}\phi-\frac{1}{6}\eta^{\mu\nu}(\partial_{\rho}\phi\partial^{\rho}\phi+m_{\phi}^{2}\phi^{2}))i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\mu}}\psi$
$\displaystyle-(\partial^{\mu}\phi\partial^{\nu}\phi-\frac{1}{4}\eta^{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi)F_{\mu\lambda}F^{\nu\lambda}]\,.$
(23)

In the above calculation, we have ignored terms in $T^{\psi}_{\mu\nu}$ that
vanish under the EoM. Matching with the effective Lagrangian, we find

$\mathcal{C}_{1}=\frac{1}{24\Lambda_{KK}^{2}m_{KK}^{2}}\,,\quad\mathcal{C}_{2}=\frac{1}{2\Lambda_{KK}^{2}m_{KK}^{2}}\,,\quad\mathcal{C}_{3}=0\,,\quad\mathcal{C}_{4}=-\frac{1}{2\Lambda_{KK}^{2}m_{KK}^{2}}\,.$
(24)

We see that in this specific example, the UV scale is the hybrid scale
$\sqrt{\Lambda_{KK}m_{KK}}$, which could be the geometrical mean of the extra-
dimension’s volume and fundamental scale in flat extra-dimension models or
just the scale of CFT breaking or confinement in RS and composite models
respectively.

As a final comment, we note that in models of spontaneously broken CFTs a
similar coupling of the form of $\partial^{\mu}\phi\partial^{\nu}\phi\times
T_{\mu\nu}/f_{\rm CFT}^{4}$, where $\phi$ is the dilaton field, $f_{\rm CFT}$
being the scale of spontaneous CFT breaking, is predicted to be present with
the coefficient of this term predicted by the $a$-anomaly Csaki:2022htl ;
Komargodski:2011vj .

###  2.4 Positivity bounds

The Wilson coefficients of the effective operators are constrained by
fundamental properties of the UV theory, such as Lorentz invariance,
unitarity, causality and locality. These properties ensure that the S-matrix
is an analytic function of the Mandelstam variables $(s,t,u)$, other than
poles and cuts determined by on-shell unitarity. The simplest bounds are
obtained by considering the two-to-two scattering amplitude $\mathcal{A}(s)$
in the forward limit in the complex plane of the $s$ (see e.g. Adams:2006sv ;
Remmen:2019cyz ). The Wilson coefficient is related to a contour integral of
the amplitude around the origin, $\oint\frac{ds}{2\pi
i}\frac{\mathcal{A}(s)}{s^{3}}$. By distorting the contour, this integral
becomes the integral of the discontinuity along the real axis up to infinity
and unitarity, which relates the discontinuity to the cross section, ensures
its positivity. In the end, this positivity bound is expressed by

$\frac{\rm d^{2}\mathcal{A}(s)}{\rm ds^{2}}\geq 0.$ (25)

To apply this to our case, we consider the contribution of the effective
operators to the forward scattering of $\phi+\psi\rightarrow\phi+\psi$ and
$\phi+\gamma\rightarrow\phi+\gamma$, in which case we obtain

$\frac{\rm d^{2}\mathcal{A}(\phi+\psi\rightarrow\phi+\psi)}{\rm
ds^{2}}=2\mathcal{C}_{2}\,,\quad\frac{\rm
d^{2}\mathcal{A}(\phi+\gamma\rightarrow\phi+\gamma)}{\rm
ds^{2}}=-\mathcal{C}_{4}/2.$ (26)

The positivity bounds thus require

$\mathcal{C}_{2}\geq 0\,,\qquad\mathcal{C}_{4}\leq 0.$ (27)

Notice that the operators $\mathcal{O}_{1}$ and $\mathcal{O}_{3}$ do not have
contributions of order $s^{2}$ to the scattering amplitude and are thus not
constrained by the positivity bound. The bounds above are consistent with the
Wilson coefficients obtained from integrating out the KK graviton, see Eq.
(24).

##  3 Experimental bounds on the operators

In this section we study the experimental implications of our EFT, assuming
that $\phi$ is a viable ultralight, sub-eV, dark matter candidate. In the rest
frame of the solar system, the field $\phi$ can be approximated as a
background field,

$\phi(t,\vec{x})=\phi_{0}\text{ cos}(m_{\phi}(t-\vec{v}_{\rm
DM}\cdot\vec{x}))\,,$ (28)

where $|\vec{v}_{\rm DM}|\simeq 10^{-3}$ Sch_nrich_2010 ; Eilers_2019  is the
relative velocity of the DM wind to the sun. The amplitude $\phi_{0}$ can be
determined by noticing that the energy density of the $\phi$ field is

$\displaystyle\rho_{\phi}=\frac{1}{2}(\dot{\phi}^{2}+m_{\phi}^{2}\phi^{2})=\frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}\,.$
(29)

Identifying this with the local DM density, $\rho_{\phi}=\rho_{\rm DM}\simeq
0.4$ GeV/cm3 Catena2010 , we have $\phi_{0}=\sqrt{2\rho_{\rm DM}}/m_{\phi}$.
Notice that for simplicity we neglect here the stochastic nature of the DM
field Centers:2019dyn ; Lisanti:2021vij ; Foster:2017hbq , as well as the DM
velocity dispersion, as these effects lead to subdominant effects below.

In cases where the spatial dependence of Eq. (28) can be neglected, the
effective operators induce time oscillations in fundamental constants such as
the speed of light, the electron mass, and the fine structure constant. These
effects are the subject of Section 3.1. In experiments testing for directional
Lorentz violation, on the other hand, the spatial dependence is crucial and
the $j=2$ operators lead to the effects discussed in Section 3.2.

Current experiments probing both kinds of effects have produced null results
and can be used to set bounds on the Wilson coefficients as described below.
The characteristic energy scales of such experiments are at or below the
electron mass, so the bounds can be interpreted as being on
$\mathcal{C}_{i}(m_{e})$ – they can be turned into bounds on Wilson
coefficients evaluated at high-energy scales of potential UV completions using
Eq. (9).

###  3.1 Bounds from the oscillation of fundamental constants

We start by noticing that, by replacing $\phi$ with its background value in
Eq. (28), $\mathcal{O}_{1}$ and $\mathcal{O}_{3}$ lead to corrections to the
electron mass term and photon kinetic term respectively 333The shift symmetry
breaking operators
$\mathcal{Q}_{1}=m_{\phi}^{2}\phi^{2}i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\nu}}\psi$
and $\mathcal{Q}_{3}=\frac{1}{2}m_{\phi}^{2}\phi^{2}F_{\nu\rho}F^{\nu\rho}$,
with coefficients $d_{1}$ and $d_{2}$, can be taken into account by
interpreting the bounds derived for $\mathcal{C}_{1}$ and $\mathcal{C}_{3}$ as
being on the combinations of them.

$\displaystyle\mathcal{C}_{1}\mathcal{O}_{1}$
$\displaystyle\longrightarrow\mathcal{C}_{1}\rho_{\rm
DM}(1-\cos(2m_{\phi}t))m_{e}\bar{\psi}\psi\,,$ (30)
$\displaystyle\mathcal{C}_{3}\mathcal{O}_{3}$
$\displaystyle\longrightarrow\frac{1}{2}\mathcal{C}_{3}\rho_{\rm
DM}(1-\cos(2m_{\phi}t))F_{\mu\nu}F^{\mu\nu}.$ (31)

In deriving these equations we have ignored the spatial dependence inside the
cosine, since the wavelength of the $\phi$ field is always much larger than
the typical size of the experiments we consider in the following, and also
used the EoM, $i\not{D}\psi=m_{e}\psi$. We find that $\mathcal{O}_{1}$ induces
a shift in the electron mass, while $\mathcal{O}_{3}$ induces a shift of the
fine structure constant (after a field redefinition to bring the photon
kinetic term into the canonical form), both oscillate with frequency
$2m_{\phi}$:

$\displaystyle\frac{\Delta m_{e}}{m_{e}}$
$\displaystyle=-\mathcal{C}_{1}\rho_{\rm DM}(1-\cos(2m_{\phi}t))\,,$ (32)
$\displaystyle\frac{\Delta\alpha}{\alpha}$
$\displaystyle=2\mathcal{C}_{3}\rho_{\rm DM}(1-\cos(2m_{\phi}t))\,.$ (33)

Next we consider the effects of $\mathcal{O}_{2}$ and $\mathcal{O}_{4}$. With
the DM background in Eq. (28) (ignoring again the spatial dependence), we have

$\displaystyle\mathcal{C}_{2}\mathcal{O}_{2}$ $\displaystyle\longrightarrow
iC_{\mu\nu}\bar{\psi}\gamma^{\mu}\overset{\leftrightarrow}{D^{\nu}}\psi\,,$
(34) $\displaystyle\mathcal{C}_{4}\mathcal{O}_{4}$
$\displaystyle\longrightarrow-\frac{1}{2}K_{\mu\nu}F^{\mu\rho}{F^{\nu}}_{\rho},$
(35)

with

$\displaystyle C_{\mu\nu}$ $\displaystyle=\mathcal{C}_{2}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\times{\rm Diag}\\{3/2,1/2,1/2,1/2\\}\,,$ $\displaystyle
K_{\mu\nu}$ $\displaystyle=-\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\times{\rm Diag}\\{3/2,1/2,1/2,1/2\\}\,.$ (36)

We see that $\mathcal{O}_{2}$ and $\mathcal{O}_{4}$ modify the kinetic terms
of the electron and photon fields in a way that discriminates between time and
spatial components, and thus their effects can not simply be taken as a shift
to fundamental constants.

In low-energy atomic physics experiments the operator $\mathcal{O}_{2}$
induces non-trivial changes to the non-relativistic QED Hamiltonian, $h$. For
instance, the kinetic term for the electron is shifted to

$\displaystyle h\supset-\left(1-\frac{5}{2}\mathcal{C}_{2}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\right)\frac{\nabla^{2}}{2m_{e}}.$ (37)

This leads to a change in the Rydberg energy, which can be obtained by
replacing $m_{e}\to(1+\frac{5}{2}\mathcal{C}_{2}\rho_{\rm
DM}\sin^{2}(m_{\phi}t))m_{e}$. However, this simple replacement is not
generally true for higher-order terms responsible for (hyper)fine structure,
such as spin-orbit and spin-spin interactions. We discuss the shifts in these
terms in Appendix C.

To see the effect of $\mathcal{O}_{4}$, we write the photon Lagrangian in the
following form:

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathcal{C}_{4}\mathcal{O}_{4}$
(38) $\displaystyle\to$ $\displaystyle\frac{1}{2}(1-\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t))E^{2}-\frac{1}{2}(1+\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t))B^{2}\dots\,,$ (39)

where the $\dots$ refers to terms suppressed by $\vec{v}_{\rm DM}$, which are
the topic of section 3.2. We see that the DM background acts as a medium with
effective permittivity $\epsilon$ and permeability $\mu$ given by

$\displaystyle\epsilon=\frac{1}{\mu}=1-\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\,.$ (40)

This implies that the speed of light is modified to

$\displaystyle c^{\prime}$ $\displaystyle=\frac{1}{\sqrt{\epsilon\mu}}\simeq
1+\mathcal{C}_{4}\rho_{\rm DM}\sin^{2}(m_{\phi}t).$ (41)

The requirement of $c^{\prime}\leq 1$ gives $\mathcal{C}_{4}\leq 0$, in
agreement with the positivity bound in Eq. (27). The coefficient
$\mathcal{C}_{4}$ also affects atomic-physics observables such as the Rydberg
energy and (hyper)fine structure – the relevant calculations are given in
Appendix C.

We now consider experimental probes of the effects discussed above.
Oscillations of the electron mass and fine structure constant due to
$\mathcal{O}_{1}$ and $\mathcal{O}_{3}$ can be probed by comparing two systems
which have typical frequencies $f_{1}$ and $f_{2}$ with different dependence
on $m_{e}$ and $\alpha$, and looking for the sinusoidal modulations of the
ratio of the two frequencies. Currently the strongest bounds on the
oscillation of $\alpha$ come from optical clock comparisons. In Lange:2020cul
; Filzinger:2023zrs , the ${}^{2}S_{1/2}(F=0)\rightarrow$ ${}^{2}D_{3/2}(F=2)$
electric quadrupole (E2) and the ${}^{2}S_{1/2}(F=0)\rightarrow$
${}^{2}F_{7/2}(F=3)$ electric octopole (E3) transitions of 171Yb+ are measured
and compared. The strong bounds benefit from the strong $\alpha$ dependence
$\frac{\Delta(f_{E3}/f_{E2})}{f_{E3}/f_{E2}}=-6.95\frac{\Delta\alpha}{\alpha}$.
Also studied in Filzinger:2023zrs , as well as in Sherrill:2023zah , is the
comparison between the E3 transition with the ${}^{1}S_{0}\rightarrow$
${}^{3}P_{0}$ transition in 87Sr, which has the sensitivity
$\frac{\Delta(f_{E3}/f_{Sr})}{f_{E3}/f_{Sr}}=-6.01\frac{\Delta\alpha}{\alpha}$.
Considering the scenario where the oscillation of $\alpha$ is due to a dark
matter background $\varphi$, with interaction of the form $\mathcal{L}\supset
d_{e}\sqrt{4\pi G}\varphi F_{\mu\nu}F^{\mu\nu}/4$ where $G$ is the Newton’s
constant, bounds are reported on the parameter $d_{e}$ in Filzinger:2023zrs ;
Sherrill:2023zah . By identifying the same amplitude of oscillation of
$\alpha$ at the same frequency, we can relate the bound on our Wilson
coefficient at given $m_{\phi}$ to the bound on $d_{e}$ at $2m_{\phi}$, with
the correspondence $\mathcal{C}_{3}(m_{\phi})\leftrightarrow\sqrt{\frac{2\pi
G}{\rho_{\rm DM}}}\frac{d_{e}(2m_{\phi})}{2m_{\phi}}$. We show the bounds on
$\mathcal{C}_{3}$ from these two experiments in the upper left plot in Fig. 2.
We see that $\mathcal{C}_{3}$ is more constrained by the Yb+/Sr comparison at
$m_{\phi}\gtrsim 10^{-18}$ eV, which requires roughly

$|\mathcal{C}_{3}|\lesssim 10^{-14}\,{\rm eV}^{-4}.$ (42)

In the upper right plot of Fig. 2 we show the corresponding bound on
$\Lambda$, using Eq. (4). The bound is very weak, $\Lambda\gtrsim 10$ keV,
compared to the scale the UV completions considered in Sec. 2.3.

The bounds on the oscillation of $m_{e}$ are generally weaker than those on
$\alpha$. In Kennedy2020 , the resonant frequency $f_{c}$ of a silicon optical
cavity and the frequency $f_{H}$ of the 1H hyperfine transition are compared,
yielding sensitivity to oscillations of $m_{e}$ according to
$\frac{\Delta(f_{H}/f_{c})}{f_{H}/f_{c}}=\frac{\Delta m_{e}}{m_{e}}$. The
bound is reported in terms of the parameter $d_{me}$, defined by the
interaction $\mathcal{L}\supset-d_{me}\sqrt{4\pi
G}m_{e}\varphi\bar{\psi}\psi$. At lower masses $m_{\phi}\lesssim 10^{-20}$ eV,
stronger bounds are provided in Kobayashi:2022vsf  by the comparison of 171Yb
optical lattice clock and a 133Cs fountain clock. Using the correspondence
$\mathcal{C}_{1}(m_{\phi})\leftrightarrow\sqrt{\frac{2\pi G}{\rho_{\rm
DM}}}\frac{d_{m_{e}}(2m_{\phi})}{m_{\phi}}$, we can translate the bounds
therein to bounds on $\mathcal{C}_{1}$. As shown in the lower left plot of
Fig. 2, we find roughly

$|\mathcal{C}_{1}|\lesssim 10^{-10}\,{\rm eV}^{-4}\,.$ (43)

The corresponding bound on the cut-off
$\Lambda=\sqrt{4\pi}|\mathcal{C}_{1}|^{-1/4}$ is shown in the lower right
plot, leading to $\Lambda\gtrsim$ keV.

There is one subtlety in re-interpreting bounds derived from linear DM
coupling as those on quadratic coupling, as we did above. When the quadratic
coupling is positive and sufficiently strong, the DM profile is modified
significantly due to interactions with the earth and the $\phi$ field gets
screened at the surface of the earth Banerjee:2022sqg ; Hees:2018fpg . Any
terrestrial experiments will thus not feel it. For our case, this happens at
$\mathcal{C}_{1,3}\gtrsim 10^{-3}(\frac{10^{-20}{\rm eV}}{m_{\phi}})^{2}{\rm
eV}^{-4}$, so is not relevant for the parameter space studied above, where the
clock experiments are most sensitive. However, at larger $m_{\phi}$ the
screening effect could be important.

In Fig. 2 we also show the bounds from equivalence principle tests. The
stringent limit is from the MICROSCOPE experiment Berge:2017ovy  which
measures the differential accelerations of macroscopic masses. We notice that
for $\mathcal{C}_{3}$ this bound is weaker than those from clocks, while for
$\mathcal{C}_{1}$ it could be stronger at $m_{\phi}\gtrsim 10^{-19}$ eV.
However, the bounds from clocks are expected to be improved significantly by
future experiments. In particular, the nuclear clock Flambaum:2006ak ;
Peik:2020cwm  is expected to improve the ability to probe scalar dark matter
by 5-6 orders of magnitude, due to its strongly enhanced sensitivity.

Figure 2: The bounds on Wilson coefficients from experiments searching for
oscillations of fundamental constants for different dark matter mass
$m_{\phi}$. Upper: Exclusion regions of $\mathcal{C}_{3}$ (left) and the
corresponding cut-off $\Lambda\sim\sqrt{4\pi}\mathcal{C}_{3}^{-1/4}$ (right).
Orange regions are obtained from Filzinger:2023zrs  (comparison of Yb+ E3/E2
as well as Yb+/Sr) and blue regions from Sherrill:2023zah  (comparison of
Yb+/Sr). Lower: Exclusion regions of $\mathcal{C}_{1}$ (left) and the
corresponding cut-off $\Lambda\sim\sqrt{4\pi}\mathcal{C}_{1}^{-1/4}$ (right),
obtained by comparing the frequencies of Yb/Cs Kobayashi:2022vsf  (orange) or
H/Si Kennedy2020  (blue). Also shown with gray shaded regions are bounds from
the MICROSCOPE experiment, obtained from direct translation of the bounds
given in Banerjee:2022sqg ; Hees:2018fpg .

Although the operators $\mathcal{O}_{2}$ and $\mathcal{O}_{4}$ lead to more
complicated effects than the simple oscillation of fundamental constants, they
can still affect the results of aforementioned frequency comparison
experiments. Consider for example the $f_{c}/f_{H}$ comparison. The frequency
of the cavity is determined by $f_{c}=Nc/2d$, where $N$ is an integer number,
$c$ is the speed of light and modified by $\mathcal{C}_{4}$ according Eq.
(41), $d$ is the cavity size and proportional to the Bohr radius $d\propto
1/m_{e}e^{2}$ which is modified by
$m_{e}\rightarrow(1+\frac{5}{2}\mathcal{C}_{2}\rho_{\rm
DM}\sin^{2}(m_{\phi}t))m_{e}$ and $e^{2}\rightarrow(1+\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t))e^{2}$ (from the correction to the Coulomb potential
due to the change in permittivity). We find that
$f_{c}\rightarrow(1+(\frac{5}{2}\mathcal{C}_{2}+2\mathcal{C}_{4})\rho_{\rm
DM}\sin^{2}(m_{\phi}t))f_{c}$. We also derive the frequency of Hydrogen
hyperfine splitting from spin-spin interaction in Appendix C and obtain
$f_{H}\rightarrow(1+(5\mathcal{C}_{2}+2\mathcal{C}_{4})\rho_{\rm
DM}\sin^{2}(m_{\phi}t))f_{H}$. In the end we have
$\frac{\Delta(f_{H}/f_{c})}{f_{H}/f_{c}}=\frac{5}{4}\mathcal{C}_{2}\rho_{\rm
DM}(1-\cos(2m_{\phi}t))$. The bound obtained for $\mathcal{C}_{1}$ from this
experiment should thus be taken as the bound for
$\mathcal{C}_{1}-5\mathcal{C}_{2}/4$. In principle, one could also derive
similar bounds from other experiments mentioned above. However, the dependence
of energy levels on $\mathcal{C}_{2}$ and $\mathcal{C}_{4}$ in high $Z$ atoms
such as 133Cs is not as simple as for the Hydrogen atom and requires numerical
calculations taking into account the many-body effects and higher order
relativistic corrections, which are beyond the scope of this work.

Concerning the correction to the speed of light in Eq. (41), we may directly
put bounds on $\mathcal{C}_{4}$ from experiments measuring the propagation of
light, for instance, the Lunar Laser Ranging experiment, where the time of
travel for a round-way trip of the laser between the earth and the moon was
measured. The oscillation in $c$ maps onto an oscillation in the time of
travel, which would be interpreted as a correction to the distance $r$ between
earth and the moon, $\Delta r\approx-r\mathcal{C}_{4}\rho_{\rm DM}\text{
sin}^{2}(m_{\phi}t)$. However, one should bear in mind that the frequency of
the clock used in the experiment would also be modified by $\mathcal{C}_{4}$.
Nevertheless, supposing that there is no non-trivial cancellation between the
two effects, we can make an estimation of the bound. In Zhang:2020wpj , the
bound of the amplitude $A$ for the oscillation of $\Delta r$ with period of
one sidereal month was obtained to be $A\lesssim 1$ mm. This gives a bound
$|\mathcal{C}_{4}|\lesssim 10^{-3}$ eV-4 at $m_{\phi}\approx 10^{-21}$ eV.
Translating this to a mass scale using Eq. (4), we have
$\Lambda\sim\sqrt{4\pi}|\mathcal{C}_{4}|^{-1/4}\lesssim 20$ eV, which is very
weak compared to the bounds from clock/cavity comparison experiments. In Sec.
3.2 we will show that linear corrections in $\vec{v}_{\rm DM}$ which were
neglected in Eq. (41) do not affect round-way trip measurements and this bound
still holds. On the other hand, quadratic corrections in $\vec{v}_{\rm DM}$ do
not cancel out in a round way trip, and stronger bounds can be obtained using
interferometers.

###  3.2 Bounds from experiments measuring Lorentz violation

One feature of particular interest in our EFT is the Lorentz violating effects
(in particular the asymmetry under spatial rotations) from $\mathcal{O}_{2}$
and $\mathcal{O}_{4}$ when $\phi$ takes the background value Eq. (28) and
$\vec{v}_{\rm DM}$ is not ignored. Our EFT thus provides a concrete model to
be tested in the large number of experiments testing Lorentz violation in
recent years.

Motivated by potential Lorentz violation at the Planck scale, the general low-
energy effective theory describing the effects of spontaneous Lorentz symmetry
breaking has been developed as the Standard-Model Extension (SME)
Colladay:1996iz ; Colladay:1998fq  and has been widely used to interpret
results from experiments hunting for Lorentz violation. For comparison with
our EFT, the relevant terms in the SME Lagrangian take the form

$\mathcal{L}_{\rm SME}\supset
i(\eta_{\mu\nu}+c_{\mu\nu})\bar{\psi}\gamma^{\mu}\overset{\leftrightarrow}{D^{\nu}}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\left(k_{F}\right)_{\kappa\lambda\mu\nu}F^{\kappa\lambda}F^{\mu\nu}.$
(44)

Here the parameters $c_{\mu\nu}$ and
$\left(k_{F}\right)_{\kappa\lambda\mu\nu}$ characterize Lorentz-violating, in
the electron and photon sector respectively. They are defined in the Sun-
centred celestial equatorial frame (SCCEF) 444This frame is defined as
follows. The $Z$-axis is parallel to the rotational axis of the Earth and
directed north. The $X$-axis points from the Sun towards the vernal equinox.
The $Y$-axis completes a right-handed system. and are assumed to be constant.
Both of them are defined to be (double) traceless, $c_{\;\mu}^{\mu}=0$,
$\left(k_{F}\right)_{\quad\mu\nu}^{\mu\nu}=0$. Furthermore,
$\left(k_{F}\right)_{\kappa\lambda\mu\nu}$ has the symmetries of the Riemann
tensor.

Ten of the 19 independent coefficients in
$\left(k_{F}\right)_{\kappa\lambda\mu\nu}$ generate a birefringence effect,
which is strongly constrained by polarization measurements of photons traveled
over cosmological distances. If these ten components are zero,
$\left(k_{F}\right)_{\kappa\lambda\mu\nu}$ can be formulated as

$\left(k_{F}\right)_{\kappa\lambda\mu\nu}=\frac{1}{2}(\eta_{\kappa\mu}k_{\lambda\nu}-\eta_{\kappa\nu}k_{\lambda\mu}-\eta_{\lambda\mu}k_{\kappa\nu}+\eta_{\lambda\nu}k_{\kappa\mu}),$
(45)

with $k_{\mu\nu}\equiv\left(k_{F}\right)_{\alpha}^{\;\mu\alpha\nu}$, which is
symmetric and traceless.

Comparing with our effective operators after substituting $\phi$ with Eq. (28)
yields

$\displaystyle c_{\mu\nu}$
$\displaystyle=\mathcal{C}_{2}\rho_{\text{DM}}(v_{\mu}v_{\nu}-\eta_{\mu\nu}/4),$
$\displaystyle k_{\mu\nu}$
$\displaystyle=-\mathcal{C}_{4}\rho_{\text{DM}}(v_{\mu}v_{\nu}-\eta_{\mu\nu}/4),$
(46)

with $v_{\mu}=\left(\sqrt{1+v_{\rm DM}^{2}},\vec{v}_{\rm DM}\right)$. Unlike
in the previous subsection, here (to be compared with the bounds for SME
parameters) we ignored the oscillation and took the average
$\langle\sin^{2}(m_{\phi}t)\rangle=1/2$. This is valid in two regimes: either
when the oscillation period is much shorter than the typical time scale of the
experiment so that the oscillation is averaged out, or when the oscillation is
so slow that the amplitude does not change much during the experiment. The
intermediate region requires a dedicated analysis of the original experimental
data and we leave it for future work. In this subsection we are more focused
on the Galilean violating effects, occurring at order $|\vec{v}_{\rm
DM}|^{2}\sim 10^{-6}\,.$

One of the most precise terrestrial tests of Lorentz violation comes from the
comparison of the E3 transitions of two 171YB+ clocks Sanner2019 . These two
clocks are placed in magnetic fields with different orientations. The magnetic
fields define the quantization axis for these two ions and break the $m_{F}$
degeneracy in the ${}^{2}F_{7/2}(F=3)$ states. For both ions the
${}^{2}F_{7/2}(F=3,m_{F}=0)$ states are selected. If no Lorentz violation
exists, the frequencies of the transitions in these two clocks should be the
same. With Lorentz violation, on the other hand, the energy levels of the
clocks are modified to be different. First, the $c_{\mu\nu}$ terms directly
add to the non-relativistic Hamiltonian of the electron a spherically
asymmetric contribution Kostelecky:1999mr . Second, Lorentz violation in the
photon sector modifies the electromagnetic potential sourced by the nucleus
Bailey:2004na . These two effects are described by the effective Hamiltonian
Kostelecky:1999mr ; Hohensee:2013cya

$\delta h=-C_{0}^{(2)}\frac{p^{2}-3p_{z}^{2}}{6m_{e}},$ (47)

with $C_{0}^{(2)}=c^{\prime}_{xx}+c^{\prime}_{yy}-2c^{\prime}_{zz}$ and
$c^{\prime}_{\mu\nu}=c_{\mu\nu}-\frac{1}{2}k_{\mu\nu}$.555The dependence on
the particular combination $c^{\mu\nu}-\frac{1}{2}k^{\mu\nu}$ can be
understood by the observation that by doing a coordinate transformation
$x^{\mu}\rightarrow x^{\mu}-\frac{1}{2}k^{\mu}_{\;\nu}x^{\nu}$ Bailey:2004na ,
the Lorentz-violating terms in the photon sector can be eliminated and the
fermion sector changes as $c^{\mu\nu}\rightarrow
c^{\mu\nu}-\frac{1}{2}k^{\mu\nu}$ Altschul:2006zz . Such a transformation
corresponds to a choice of “rods and clocks” defined by the freely-propagating
electromagnetic field in the experiment.

Notice that the $c^{\prime}_{\mu\nu}$ in Eq. (47) is the value in the clock
frame while the parameters in Eq. (44) are defined in the SCCEF frame, so that
one has to do a time-dependent Lorentz transformation to relate them due to
the orbiting of the earth around the sun as well as the rotation of the earth.
The experiment thus has sensitivities not only to the spatial components
$c^{\prime}_{XY}$ but also to $c^{\prime}_{TX}$ and $c^{\prime}_{TT}$ 666Here
and in the following we use the uppercase indices for the SCCEF coordinates
and lowercase indices for coordinates in the lab frame.. However, these are
suppressed by a small boost factor, $\beta\sim 10^{-4}$ and therefore result
in weaker bounds.

With the data from a six-month measurement period, Sanner2019  reports bounds
on Lorentz-violating parameters. The strongest bounds are put on the spatial
components $c^{\prime}_{XY}\lesssim 10^{-20}$. This implies that

$\mathcal{C}_{2}+\mathcal{C}_{4}/{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}\lesssim
2\times 10^{-9}\text{ eV}^{-4}\,,$ (48)

which translates to a bound on the cutoff of $\Lambda\gtrsim 0.5$ keV.

Lorentz violation in the photon sector can also be probed with
interferometers, because the speed of light is modified and depends on the
direction of propagation. In the lab frame, we have the modification of the
effective refractive index $\bar{n}$ Kostelecky:2016kkn :

$\delta\bar{n}=-\frac{1}{2}\hat{l}^{j}\hat{l}^{k}\tilde{\kappa}^{jk}_{e-}+\frac{1}{2}\epsilon^{jkl}\hat{l}^{j}\tilde{\kappa}^{kl}_{o+}+\tilde{\kappa}_{\rm
tr},$ (49)

where $\tilde{\kappa}^{jk}_{e-}=-k^{jk}+\frac{1}{3}k_{ii}\delta^{jk}$,
$\tilde{\kappa}^{jk}_{o+}=-\epsilon^{ijk}k_{0i}$, $\tilde{\kappa}_{\rm
tr}=\frac{2}{3}k_{00}$ Kostelecky:2002hh , and $\hat{l}$ denotes the direction
of the propagation. In the interferometer experiments, two beams of light are
sent to two different directions $\hat{l}_{1}$ and $\hat{l}_{2}$ and compared
when reflected back. The effect of the second term in Eq. (49) cancels out in
this two way measurement (since $\hat{l}$ changes sign on the way back). Then
the difference between these two beams is only sensitive to
$\tilde{\kappa}^{jk}_{e-}$:

$\bar{n}_{1}-\bar{n}_{2}=\delta\bar{n}_{1}-\delta\bar{n}_{2}=-\frac{1}{2}(\hat{l}_{1}^{j}\hat{l}_{1}^{k}-\hat{l}_{2}^{j}\hat{l}_{2}^{k})\tilde{\kappa}^{jk}_{e-}.$
(50)

Again the Lorentz transformation from the lab frame to the SCCEF provides the
(weaker) sensitivity of the experiment also to other LV parameters.

In Kostelecky:2016kkn , the data from Laser Interferometer Gravitational-Wave
Observatory (LIGO) was analysed and the null LV signal 777In fact an anomaly
with semi-annual frequency is found from the analysis. However this should not
be interpreted as signal of LV since it is imcompatible with the constraints
obtained from other frequencies. gives strong bounds
$|\tilde{\kappa}^{XY}_{e-}|<2.7\times 10^{-22}$. Using Eq. (3.2) and the
relations under Eq. (49), together with the positivity bound, this implies

$-2\times 10^{-10}\text{ eV}^{-4}\lesssim\mathcal{C}_{4}\leq 0,$ (51)

or in terms of the cut-off, $\Lambda\gtrsim$ keV.

Let us comment on the range of $m_{\phi}$ for which the bounds from the two
experiments above apply. In the clock comparison experiment, the data is taken
every roughly 2 seconds and covers a period of $T_{e}\approx 4\times 10^{6}$
s. This means either $m_{\phi}\ll 10^{-21}$ eV so that the period of is
oscillation much longer than $T_{e}$, or $m_{\phi}\gg 10^{-15}$ eV so that the
oscillation is averaged out. For the analysis of data from LIGO, which covers
a period of 16 months, the slow oscillation regime corresponds to $m_{\phi}\ll
10^{-22}$ eV. For the fast oscillation regime, we require $2\pi/m_{\phi}\ll
2L/c$, where $L\simeq 4$ km is the arm length of LIGO. This corresponds to
$m_{\phi}\gg 10^{-11}$ eV. Away from these mass ranges, the oscillating nature
of $\phi$ cannot be ignored when obtaining bounds from the experimental data.

##  4 Astrophysical, cosmological and collider bounds

In addition to terrestrial experiments, our theory can also leave signals in
astrophysical and cosmological observations, warranting dedicated
investigation that we leave for future work. However, we note the bounds
derived from such observations often suffer from considerable uncertainties or
rely on specific assumptions. For instance, the observation of high-energy
cosmic rays could place strong constraints on Lorentz violation
Altschul:2021wrm ; Klinkhamer:2007ak ; HAWC:2019gui . If the coefficient
$\tilde{\kappa}_{\rm tr}$ is positive, charged particles experience rapid
energy loss through Cherenkov radiation when their energy exceeds a critical
energy $E_{c}$ determined by $\tilde{\kappa}_{\rm tr}$. Thus, observations of
charged particles with energy $E_{o}$ that have traversed astronomical
distances imply that $E_{o}<E_{c}$, leading to bounds on $\tilde{\kappa}_{\rm
tr}$. The specific bound for our case depends on the distribution as well as
stochastic fluctuations of dark matter on the larger scales between the cosmic
source and Earth.

There are additional bounds that do not assume $\phi$ as dark matter. The
interaction between SM particles and light particles with masses $\lesssim
100$ MeV can be constrained by their effect on the rate of stellar cooling
Raffelt:1996wa . As an example, consider the process of photon annihilation
into a pair of $\phi$ induced by the operator $\mathcal{O}_{3}$. The cross
section in the center of mass frame is given by
$\sigma_{\gamma\gamma\rightarrow\phi\phi}=\mathcal{C}_{3}^{2}\omega^{6}/32\pi$,
where $\omega$ is the photon energy. In the thermal bath of photons, the
energy loss rate per unit volume can be estimated as Olive:2007aj
$\Gamma_{\gamma\gamma\rightarrow\phi\phi}=n_{\gamma}^{2}\langle
2\omega\sigma_{\gamma\gamma\rightarrow\phi\phi}\rangle\approx
178\mathcal{C}_{3}^{2}T^{13}$. The constraint of the additional energy loss
(per unit volume and time) in the supernovae burst SN1987A is about
$\Gamma<10^{-14}\text{ MeV}^{5}$, with the core temperature $T\approx$ 30 MeV
Raffelt:1996wa . This gives the bound
$\Lambda\sim\sqrt{4\pi}/\mathcal{C}_{3}^{1/4}\gtrsim 100$ GeV. The cooling of
Horizontal Branch and Red Giant stars provide bounds in a similar way. With
typical core temperature $T\approx 10$ keV, they give $\Lambda\gtrsim$ GeV,
which are weaker then that from SN1987A, but still significantly stronger than
the bounds derived from terrestrial experiments. However, it is noteworthy
that all these cooling bounds can potentially be avoided through density
(Chameleon) effects DeRocco:2020xdt ; Bloch:2020uzh ; Budnik:2020nwz .

On the cosmological front, for $\phi$ being ultralight particles, they must
decouple from the thermal bath prior to neutron decoupling to avoid changing
the effective relativistic degrees of freedom and impacting the Big Bang
Nucleosynthesis (BBN) process. This requirement leads to the condition
$n\langle\sigma v\rangle<H$, where $n\sim T^{3}$ is the number density ($T$
being the temperature at decoupling), $\langle\sigma v\rangle\sim
T^{6}/\Lambda^{8}$ is the averaged cross section, and $H\sim T^{2}/M_{\rm P}$
is the Hubble parameter ($M_{\rm P}$ being the Planck mass). Consequently, we
obtain the constraint $\Lambda>(T^{7}M_{P})^{1/8}$. Considering $T\sim$ MeV,
we find $\Lambda\gtrsim$ GeV. Again, this stringent bound can potentially be
circumvented through density effects.

Collider searches also offer valuable insights into our model. For instance,
we can utilize data from the Large Electron-Positron Collider (LEP) on mono-
photon events with significant missing energy to constrain the coupling of
dark matter to electrons. The SM background in such cases is
$e^{+}e^{-}\rightarrow\gamma\nu\nu$, and it is found to be in agreement with
measurements (see Table 6 in L3:2003yon ). By imposing the requirement that
the additional cross-section from $e^{+}e^{-}\rightarrow\gamma\phi\phi$ is
smaller than the uncertainty, we can constrain the couplings $\mathcal{C}_{1}$
and $\mathcal{C}_{2}$. We do not consider such bounds in this work,
considering the possibility that $\phi$ could be a composite state, which
should be replaced by more fundamental degrees of freedom at collider
energies.

##  5 Discussion and conclusions

In this paper we have studied an effective theory of scalar ultralight dark
matter (ULDM) with quadratic couplings to the QED sector of the SM. Assuming a
shift and $Z_{2}$ symmetry, the leading interactions between the ULDM and the
SM arise from dimension-8 operators. We calculated the one-loop RG mixing of
such operators and showed that to all orders they can be can be classified
into two sectors, $j=0$ or $j=2$, without any RG mixing between them. Such
operators can be associated with a variety of fundamental theories and in some
cases may be the leading interactions between the ULDM and the SM – as an
explicit example we showed how they can be generated by massive spin-2
graviton exchange. The ULDM background leads to Lorentz and Galilean violating
effects, which we shall refer to collectively as Lorentz violating (LV).

The sign of the Wilson coefficients ($\mathcal{C}$) in the $j=2$ sector is
constrained by positivity bounds. Regarding experimental tests of our theory,
we considered two types of terrestrial experiments: those investigating the
oscillation of fundamental constants and those searching for Lorentz-violating
effects. We found that the first kind of experiment provides bounds on the
Wilson coefficients of both sectors, while only the operators of $j=2$ sector
are probed in the second kind of experiment. When the bounds are translated to
a mass scale using $\mathcal{C}\sim 16\pi^{2}/\Lambda^{4}$, the current
sensitivity of these two kinds of measurements only reaches the scale of
$\Lambda\gtrsim\mathcal{O}(1-10)$ keV. These bounds are found to be rather
weak, because the effects of the dimension-8 operators in terrestrial
experiments are suppressed by $\rho_{\rm
DM}/\Lambda^{4}\approx(0.04\rm{\,eV}/\Lambda)^{4}$. We note that this scaling
is not unique to our model, but generally exists for other Lorentz-violating
ULDM models. For instance, consider a dark photon $A^{\prime}_{\mu}$. To
obtain Galilean violation one must replace
$\partial_{\mu}\phi\partial_{\nu}\phi\rightarrow
A^{\prime}_{\mu}A^{\prime}_{\nu}$ in the operators discussed above. This would
seem to generate dimension-6 operators, but there is an extra suppression of
$m_{A}^{2}/\Lambda^{2}$ due to U(1) gauge symmetry. The dipole operator –
which is dimension 5 – leads to magnetic-like DM background that interacts
with electron spin, and generically cannot be probed in the clock comparison
experiments.

The bounds obtained here could be improved by an order of magnitude with
future clock experiments. Moreover, the potential existence of an ULDM solar
halo Banerjee:2019xuy ; Banerjee:2019epw ; Budker:2023sex , leading to a bound
DM density in the solar system approximately $10^{4}$ times larger than that
of the standard halo model, would also elevate the bounds by a factor of 10.

Unlike the approach commonly discussed thus far in the literature (based on
the SME ansatz), which is not based on a consistent effective theory, our EFT
of ULDM gives a consistent description of Lorentz and Galilean symmetry
violation. It leads to several other interesting implications, namely that we
can consistently compare the bounds associated with Galilean symmetry
violation to those associated with oscillation of fundamental constants and
tests of equivalent-principle violation. Interestingly, enough, we find that
these completely different three classes of experiments currently probe
roughly the keV scale. In addition, we have found that stronger but indirect
bounds arise from astrophysical and cosmological observations, which probe the
0.1-100 GeV scale. However, these bounds are model dependent, and can be
avoided by density and temperature effects which may give rise to dramatic
changes in the ULDM mass and coupling, thus inhibiting its production in stars
or in the early universe.

###### Acknowledgements.

The authors thank Marco Gorghetto, Wei Xue, Marianna Safronova and Hyungjin
Kim for valuable discussions. The work of MJ is supported by a research grant
from the Musk Foundation. The work of GP is supported by grants from BSF-NSF,
Friedrich Wilhelm Bessel research award of the Alexander von Humboldt
Foundation, GIF, ISF, Minerva, SABRA - Yeda-Sela - WRC Program, the Estate of
Emile Mimran, and the Maurice and Vivienne Wohl Endowment. BP is grateful to
the Weizmann Institute of Science for its kind hospitality and support through
the SRITP and the Benoziyo Endowment Fund for the Advancement of Science.

##  Appendix A Eliminating the $\overline{O}_{1}$ operator

We can use the following field redefinition to absorb $\overline{O}_{1}$ into
$O_{1}$

$\psi\longrightarrow\psi+\frac{\overline{c}_{1}}{m_{\psi}}\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\psi\,,$
(52)
$\bar{\psi}\longrightarrow\bar{\psi}+\frac{\overline{c}_{1}}{m_{\psi}}\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\bar{\psi}\,.$
(53)

Under the above redefinition,

$i\bar{\psi}\overset{\leftrightarrow}{\not{D}}\psi\longrightarrow
i\bar{\psi}\overset{\leftrightarrow}{\not{D}}\psi+\frac{\overline{c}_{1}}{m_{\psi}}O_{1}\,,$
(54) $-m_{\psi}\bar{\psi}\psi\longrightarrow-
m_{\psi}\bar{\psi}\psi-\overline{c}_{1}\overline{O}_{1}\,,$ (55)

where we ignored terms of higher dimension operators ($d>10$). Thus, the
Lagrangian now becomes

$\mathcal{L}\longrightarrow\mathcal{L}_{\text{QED}}+\left(c_{1}+\frac{\overline{c}_{1}}{m_{\psi}}\right)O_{1}+\sum_{i=2}^{4}c_{i}O_{i}\,.$
(56)

We can now redefine the coefficient $c_{1}\longrightarrow
c_{1}-\frac{\overline{c}_{1}}{m_{\psi}}$, which leads to the Lagrangian in Eq.
(3).

##  Appendix B RG flow calculation

###  B.1 New vertices

Apart from the QED vertex, our calculation involves six new vertices
associated with the dimension-8 operators. Consider the expanded form of
operators $O_{1}$ and $O_{3}$:

$\displaystyle O_{1}$
$\displaystyle=\partial_{\mu}\phi\partial^{\mu}\phi\,i\bar{\psi}\gamma^{\nu}\overset{\leftrightarrow}{D_{\nu}}\psi$
$\displaystyle=\underbrace{\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\,i\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\,i(\partial_{\nu}\bar{\psi})\gamma^{\nu}\psi}_{\text{V1}}-\underbrace{e\partial_{\mu}\phi\partial^{\mu}\phi\,A_{\nu}\bar{\psi}\gamma^{\nu}\psi\,,}_{\text{V3}}$
$\displaystyle O_{3}$
$\displaystyle=\underbrace{\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\,F_{\nu\rho}F^{\nu\rho}\,,}_{\text{V2}}$

The terms V1, V2 and V3 correspond to the following vertices,

Figure 3: New vertices

Operators $O_{2}$ and $O_{4}$ can be expanded similarly and they contain three
more vertices that look identical to the above.

###  B.2 Diagrams contributing to operator mixing at one-loop order

The RG equation for the Wilson coefficients follows from the structure of the
operator renormalization matrix for $O_{1}$ – $O_{4}$. Operator
renormalization in the $\overline{\hbox{MS}}$ scheme can be extracted from the
UV divergent parts of photon-DM scattering and electron-DM scattering. There
are several diagrams contributing to each of these processes, which are shown
below. Evaluating these diagrams and requiring that the scattering amplitudes
are UV finite after including wavefunction renormalization of the electron and
photon fields leads to Eq. (5).

####  B.2.1 First set of diagrams (electron-DM scattering)

The relevant diagrams for electron-DM scattering are given in Figure 4.

Figure 4: Electron-DM scattering diagrams

####  B.2.2 Second set of diagrams (photon-DM scattering)

The relevant diagrams for photon-DM scattering are given in Figure 5.

Figure 5: Photon-DM scattering diagrams

##  Appendix C Energy levels of the Hydrogen atom

In this section we give results for the dependence of the hydrogen atom energy
levels on the Wilson coefficients $\mathcal{C}_{2}$ and $\mathcal{C}_{4}$,
when the field $\phi$ takes on the background value in Eq. (28). To derive
them, we have obtained the total Hamiltonian for the electron-proton system
interacting with the electromagnetic field in the non-relativistic limit using
the standard techniques outlined in, for instance, Berestetskii:1982qgu . This
involves taking the non-relativistic limit of the free Dirac Lagrangian, and
then calculating the interaction Hamiltonian from the Fourier transform of the
Born-level electron-proton scattering amplitude in the same limit. In
presenting the results, it is useful to introduce the quantities

$\displaystyle\tilde{c}\equiv\mathcal{C}_{2}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\,,\quad\tilde{k}\equiv\mathcal{C}_{4}\rho_{\rm
DM}\sin^{2}(m_{\phi}t)\,.$ (57)

Time derivatives acting on these objects are suppressed by $m_{\phi}$ and are
thus neglected in what follows.

Focusing on the terms relevant for the Rydberg energy, the spin-orbit
interaction, and the spin-spin interaction, the resulting time-dependent
Schrodinger equation takes the form

$\displaystyle\left(1+\frac{3}{2}\tilde{c}\right)i\partial_{t}\psi=\left(\tilde{H}_{0}+\tilde{H}_{s-o}+\tilde{H}_{s-s}\right)\psi\,.$
(58)

The Rydberg energy is obtained by setting the spin-orbit ($\tilde{H}_{s-o}$)
and spin-spin ($\tilde{H}_{s-s}$) Hamiltonians to zero, and solving the energy
levels using the leading-order Hamiltonian

$\displaystyle\tilde{H}_{0}=-e\left(1+\frac{3}{2}\tilde{c}+\tilde{k}\right)A_{0}-\left(1-\tilde{c}\right)\frac{\nabla^{2}}{2m_{e}}\,,$
(59)

with $A_{0}=e/(4\pi r)$ the Coulomb potential. Explicitly, one finds the
eigenvalue equation

$\left(-\frac{\tilde{\alpha}}{r}-\frac{\nabla^{2}}{2\tilde{m}_{e}}\right)\psi_{nlm}=\tilde{E}_{n}\psi_{nlm},$
(60)

where we have defined

$\displaystyle\tilde{\alpha}=\left(1+\tilde{k}\right)\alpha\,,\quad\tilde{m}_{e}=\left(1+\frac{5}{2}\tilde{c}\right)m_{e}\,.$
(61)

It follows that

$\displaystyle\frac{\tilde{E}_{n}}{E_{n}}=\frac{\tilde{m}_{e}\tilde{\alpha}^{2}}{m_{e}\alpha^{2}}=1+\frac{5}{2}\tilde{c}+2\tilde{k}$
(62)

where $E_{n}$ are the standard energy levels of the hydrogen atom obtained by
setting the Wilson coefficients set to zero.

We next turn to spin-orbit and spin-spin splittings, which are treated in
perturbation theory. To this end, for a given value of $nlm$, we first write
the energy level $E$ in presence of these perturbations as

$\displaystyle\tilde{E}=\tilde{E}_{n}+\Delta\tilde{E}_{s-o}+\Delta\tilde{E}_{s-s}\,.$
(63)

For the spin-orbit Hamiltonian, we find

$\displaystyle\tilde{H}_{s-o}=\left(1+\frac{3}{2}\tilde{c}\right)\frac{\tilde{\alpha}}{4\tilde{m}_{e}}\left(1+4\tilde{c}\right)\vec{\sigma_{e}}\cdot\left(\vec{\nabla}\frac{1}{r}\times
i\vec{\nabla}\right)\,,$ (64)

where $\vec{\sigma}_{e}$ is the electron spin operator. From the fact that
$\Delta E_{s-o}/E_{n}\propto\alpha^{2}$, it is easy to show that

$\displaystyle\frac{\Delta\tilde{E}_{s-o}}{\tilde{E}_{n}}=\left(1+4\tilde{c}+2\tilde{k}\right)\frac{\Delta
E_{s-o}}{E_{n}}\,.$ (65)

Finally, for the spin-spin Hamiltonian, we find

$\displaystyle\tilde{H}_{s-s}=\frac{e}{2m_{e}}\left(1-\tilde{c}\right)\times\left(1-\tilde{k}\right)\vec{\sigma}_{e}\cdot\vec{B}\,,$
(66)

where

$\displaystyle\vec{B}=\frac{g_{p}e}{2m_{p}}\left[\frac{1}{4\pi
r^{3}}[(3\vec{\sigma_{p}}\cdot\hat{r})\hat{r}-\vec{\sigma}_{p}]+\frac{2}{3}\vec{\sigma}_{p}\delta^{3}(\vec{r})\right]\,.$
(67)

Here $\vec{\sigma}_{p}$ and $m_{p}$ are the spin operator and mass of the
proton, respectively, while $g_{p}$ is the $g$-factor parameterizing its
magnetic moment. Proceeding in analogy with the spin-orbit term, we then find

$\displaystyle\frac{\Delta\tilde{E}_{\rm s-s}}{\tilde{E}_{n}}$
$\displaystyle=\left(1+\frac{5}{2}\tilde{c}\right)\frac{\Delta E_{\rm
s-s}}{E_{n}}\,.$ (68)

The dependence of the energy levels on the Wilson coefficients as given above
is a non-trivial result of the calculation. However, it can be understood
heuristically starting from the relevant terms in the standard NRQED
Lagrangian (see CASWELL1986437 ):

$\displaystyle{\cal L}_{\rm
NRQED}=\psi_{e}^{\dagger}\left(iD^{0}+\frac{\vec{D}^{2}}{2m_{e}}-\frac{e}{2m_{e}}\vec{\sigma}\cdot\vec{B}-\frac{e}{8m_{e}^{2}}\vec{\sigma}\cdot\left[i\vec{D}\times\vec{E}-\vec{E}\times
i\vec{D}\right]\right)\psi_{e}\,,$ (69)

where $\psi_{e}$ is the non-relativistic electron field. Note that, according
to Eq. (34), the effect of $\tilde{c}$ is to modify the time and spatial
components of the covariant derivatives by

$\displaystyle D^{0}\rightarrow\tilde{D}^{0}=(1+3\tilde{c}/2)D^{0}\,,\quad
D^{i}\rightarrow\tilde{D}^{i}=(1-\tilde{c}/2)D^{i}\,.$ (70)

If we furthermore treat $A^{\mu}$ as an external field sourced by the proton,
with

$\displaystyle\vec{E}=\vec{\nabla}A_{0}\,,\quad\vec{B}=\vec{\nabla}\times\vec{A}$
(71)

then we can obtain the remaining dependence on the Wilson coefficients through
the modifications

$\displaystyle\vec{E}$
$\displaystyle\to\tilde{\vec{E}}=\frac{1}{\epsilon}\left(1+\tilde{c}\right)\vec{E}\,,\,$
$\displaystyle\vec{B}$
$\displaystyle\to\tilde{\vec{B}}=\mu\left(1-\tilde{c}\right)\vec{B}\,,$ (72)

where $\vec{E}=\vec{\nabla}e/(4\pi r)$ and $\vec{B}$ is given in Eq. (67). The
dependence on $\tilde{c}$ in the above equation is obtained by applying
replacements analogous to Eq. (70) on the $A^{0}$, $\nabla^{i}$ and $A^{i}$
appearing in Eq. (71), while the dependence on $\tilde{k}$ is reflected in the
change in the permittivity and permeability in the Maxwell Lagrangian as given
in Eq. (40).

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