# Probing Grand Unification with Quantum Sensors

Xavier Calmet<EMAIL_ADDRESS>Nathaniel Sherrill
<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Sussex, Brighton, BN1 9QH, United Kingdom

###### Abstract

We discuss how grand unification can be probed with experiments at low
energies using quantum sensors. Specifically, we show that scalar multiplets
coupled to the gauge sector of a grand unified theory provide a mechanism for
a time-varying unified coupling which has low-energy consequences which can be
probed with quantum sensors. We then assume that the multiplets represent
ultra light dark matter. Constraints on ultra light dark matter couplings to
regular matter are extracted using atomic clock comparisons, pulsar timing
arrays (NANOGrav) and MICROSCOPE.

##  I Introduction

The stability of fundamental constants has long been questioned starting with
the seminal work of Dirac Dirac . Since then, this topic has been revisited
frequently by famous scientists. E.g., in 1996 Dyson and Damour Damour:1996zw
studied bounds on the stability of the fine-structure constant using data from
Oklo. Some theories of quantum gravity, typically involving extra dimensions,
predict a cosmological time evolution of fundamental constants which are then
moduli, see e.g., Polchinski:1998rq ; Polchinski:1998rr ; Marciano84 ; CO94 ;
Fritzsch:2016ewd . A comprehensive review of the fundamental constants and
their variation can be found in Uzan:2002vq .

Due to recent technological progress, quantum sensors and other atomic clocks
(see e.g. Sherrill:2023zah ; Barontini:2021mvu ; Filzinger23 ; Chou:2023hcc ;
Buchmueller:2022djy ) have made impressive advances which enable to test the
stability of fundamental constants with much improved precision. These
technological advances have revitalised the field and led to new ideas of how
to probe fundamental physics using quantum sensors.

Recently, there has been significant interest in fundamental constants which
are effectively time dependent due to the presence of time-evolving scalar
fields. These scalar fields can be thought of as giving a time-dependent
contribution to the fundamental constant. This is the case, for example, with
models of ultra light dark matter DP14 ; SF14 ; AHT15 , which have been now
fairly extensively studied.

It has also been pointed out that a time variation of fundamental constants
could enable us to probe grand unified theories at very low energy without
ever producing any of the heavy particles associated with such models CF2002 .
This is because the unification mechanism imposes relations between the time
variation of, e.g., the fine-structure constant and, e.g., the proton mass. To
the best of our knowledge, there has not been any concrete proposal to
generate a time variation of the unified coupling constant within models of
grand unification.

The aim of this paper is twofold. We first provide a mechanism to explain a
time variation of the unified gauge coupling within the context of grand
unified theories. Our mechanism involves a scalar multiplet charged under the
grand unified theory group. We then assume a special case where the multiplet
is identified with ultra light dark matter. Ultra light dark matter is a topic
which has received a considerable amount of attention recently (see, e.g.,
recent reviews SMDM ). As far as we are aware, this work is the first attempt
to embed this dark-matter candidate in a grand unified theory.

This paper is organized in the following way. We first describe how a scalar
multiplet of a grand unified group can generate an effective time variation of
the unified coupling constant of this grand unification group via dimension-
five operators generated by quantum gravity. We investigate consequences for
low-energy coupling constants. We then assume that the scalar multiplet
accounts for dark matter and posit that the scalar multiplet is ultra light.
We derive limits on the coupling between the scalar multiplet and standard
model particles using data from clocks, NANOGrav and MICROSCOPE. We then
conclude.

##  II A mechanism for the time variation of fundamental constants

It was pointed out a while ago CF2002  that a time variation of fundamental
constants of the standard model would enable us to probe unification physics,
i.e., a very high-energy scale of the order of $10^{16}$ GeV with low-energy
experiments involving, e.g., atomic clocks.

An effective time variation of the unified coupling constant can be generated
by a dimension-five operator involving scalar multiplets $H$ and the field
strength of the unified theory ${\cal G}_{\mu\nu}$. Operators of the type
$\text{Tr}(H\cal{G}_{\mu\nu}\cal{G}^{\mu\nu})$ will lead to an effective time
variation of the unified coupling constant if $H$ are time-dependent classical
fields.

While the mechanism highlighted here is generic and can be embedded in any
grand unified theory, for definiteness we consider supersymmetric $SU(5)$ as a
unification group. We assume that supersymmetry is broken at some low-energy
scale and that unification takes place at some $10^{16}$ GeV. Note that a time
variation of fundamental constants due to a background field has been
considered previously in the context of axion physics HR84 ; HST90 .

Starting from the standard ${\cal N}=1$ supersymmetric $SU(5)$ grand unified
theory à la Georgi and Glashow, we consider additional dimension-five
effective operators consisting of a unified gauge field strength ${\cal
G}^{a}_{\mu\nu}$ and scalar multiplets $H^{ab}_{r}$:

${\cal L}=-\frac{1}{4}{\cal G}_{\mu\nu}^{a}{\cal
G}^{a\mu\nu}-\frac{1}{4}\sum_{r}g_{u}^{2}\frac{c_{r}}{M_{\rm
P}}H^{ab}_{r}{\cal G}_{\mu\nu}^{a}{\cal G}^{b\mu\nu},$ (1)

where $M_{\rm P}$ is the reduced Planck mass, $g_{u}$ the unified coupling
constant, and $c_{r}$ are dimensionless coefficients where $r$ denotes the
scalar multiplet representation. Quantum gravity will generically generate
dimension-five and higher operators (see, e.g., Refs. Hill84 ; SW84 ; tHooft80
). We restrict ourselves to the leading-order quantum-gravitational
corrections to the $SU(5)$ Lagrangian density. As $H^{ab}_{r}$ are charged
under $SU(5)$, the Wilson coefficients depend on the coupling $g_{u}^{2}$.

Our aim is to study the time dependence of the standard model within this
unified theory framework. The fine-structure constant, Fermi constant and the
strong-coupling constant will be time dependent if the unified coupling
constant receives time-dependent corrections. The easiest way to achieve this
is to assume that scalar multiplets $H^{ab}_{r}$ are time-dependent classical
background fields. They could be space-time dependent, but we limit attention
to their time dependence. We will first show that the dimension-five operators
lead to an effective time variation of the unified coupling constant. These
dimension-five operators thus represent a mechanism for the time variation of
the unified coupling constant posited in CF2002 .

We will assume that the scalar multiplets $H^{ab}_{r}$ can be much lighter
than the unification scale $M_{u}$. This does not lead to any issue with
typical tests of grand unified theories, such as proton decay, as we do not
couple these extra multiplets to fermions. Explaining why scalar fields are
lighter than unification scale in grand unified theories (e.g., the mass of
the Higgs boson) has been a concern for many authors. Here, we do not attempt
to justify the masses of $H^{ab}_{r}$ which we cannot a priori calculate from
first principles. As these parameters need to be renormalized, we do not
consider this a mathematically well-posed question within the realm of quantum
field theory.

Rescaling the gauge fields by $g_{u}^{-1}$ makes it easier to identify the
effective time-dependent unified coupling constant corresponding to the
canonically normalized kinetic term. We obtain

${\cal L}=-\frac{1}{4g_{u}^{2}}{\cal G}_{\mu\nu}^{a}{\cal
G}^{a\mu\nu}-\frac{1}{4}\sum_{r}\frac{c_{r}}{M_{\rm P}}H^{ab}_{r}{\cal
G}_{\mu\nu}^{a}{\cal G}^{b\mu\nu},$ (2)

and find $({\bar{g}_{u}})^{-2}=(g_{u})^{-2}+\sum_{r}(c_{r}/M_{\rm
P})H^{aa}_{r}$, where $H^{aa}_{r}=H_{r}$ is the singlet component of
$H_{r}^{ab}$. In other words, we now have a unified coupling constant

$\displaystyle\alpha_{u}(t)=\frac{{\bar{g}_{u}^{2}}}{4\pi},$ (3)

which is effectively time dependent due to $H_{r}$.

After $SU(5)$ symmetry breaking, $H_{r}$ contributes to the gauge kinetic
terms such that at the unification scale

${\cal
L}=-\frac{1}{4}\left(\frac{1}{g_{1}^{2}}+\epsilon_{1}\right)F_{\mu\nu}F^{\mu\nu}_{U(1)}-\frac{1}{2}\left(\frac{1}{g_{2}^{2}}+\epsilon_{2}\right){\rm
Tr}F_{\mu\nu}F^{\mu\nu}_{SU(2)}-\frac{1}{2}\left(\frac{1}{g_{3}^{2}}+\epsilon_{3}\right){\rm
Tr}F_{\mu\nu}F^{\mu\nu}_{SU(3)},$ (4)

where

$\epsilon_{i}=\sum_{r}\frac{c_{r}}{M_{\rm P}}H_{r}\delta_{i}^{(r)}.$ (5)

The time-dependent couplings for $U(1)_{Y},SU(2)_{L}$, and $SU(3)_{c}$ are
given in terms of the time-independent couplings $g_{i}$ and the
$\epsilon_{i}$ by

$(\bar{g}_{i})^{-2}=(g_{i})^{-2}+\epsilon_{i}.$ (6)

The Clebsch-Gordan coefficients $\delta_{i}^{(r)}$ for the gauge-invariant
irreducible representations are given in Table 1 slansky ; CHR . For $SU(5)$,
the irreducible representation $r=$ 1, 24, 75, 200 contain exactly one
standard model singlet. For other groups, e.g., $SO(10)$, there may be more
than one standard model singlet. Note for $r=\bf{1}$ the coefficients are
equal for all subgroups and $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}$.

Table 1: Clebsch-Gordan coefficients in Eq. (5). $SU(5)$ irrep. $r$ | $\delta_{1}^{(r)}$ | $\delta_{2}^{(r)}$ | $\delta_{3}^{(r)}$
---|---|---|---
1 | $-1/\sqrt{24}$ | $-1/\sqrt{24}$ | $-1/\sqrt{24}$
24 | $1/\sqrt{63}$ | $3/\sqrt{63}$ | $-2/\sqrt{63}$
75 | $5/\sqrt{72}$ | $-3/\sqrt{72}$ | $-1/\sqrt{72}$
200 | $-10/\sqrt{168}$ | $-2/\sqrt{168}$ | $-1/\sqrt{168}$

At the unification scale $M_{u}$, we have

$\alpha_{i}(M_{u})=\bar{g}^{2}_{i}(M_{u})/(4\pi)=\alpha_{u}(t,M_{u}).$ (7)

We thus have a mechanism to generate an effective time dependence of the
unified coupling constant. Note that the unification condition is time
independent and not affected by the dimension-five operators in Eq. (4).

Since $\bar{g}_{i}$ are the canonically normalized couplings, their scale
evolution at 1-loop is

$\displaystyle\frac{1}{\alpha_{i}(\mu)}=\frac{1}{\alpha_{u}}+\frac{1}{2\pi}b_{i}\ln\left(\frac{M_{u}}{\mu}\right).$
(8)

The parameters $b_{i}$ are given by
$b^{\rm{SM}}_{i}\\!\\!\\!=(b^{\rm{SM}}_{1},b^{\rm{SM}}_{2},b^{\rm{SM}}_{3})=(41/10,-19/6,-7)$
below the supersymmetric scale and by
$b^{\rm{S}}_{i}=(b^{\rm{S}}_{1},b^{\rm{S}}_{2},b^{\rm{S}}_{3})=(33/5,1,-3)$
when ${\cal N}=1$ supersymmetry is restored.

Note that the coupling constants $\alpha_{i}$ depend not only on the scale
$\mu$, but also on the time-dependent scalar fields $H_{r}$. Since the
coefficients $b_{i}$ are time independent, one finds

$\displaystyle\frac{1}{\alpha_{i}(\mu)}\frac{\dot{\alpha}_{i}(\mu)}{\alpha_{i}(\mu)}=\frac{1}{\alpha_{i}(\mu^{\prime})}\frac{\dot{\alpha}_{i}(\mu^{\prime})}{\alpha_{i}(\mu^{\prime})}\
\ \ i\in\\{1,2,3\\},$ (9)

i.e., the quantity $\alpha_{i}^{-1}(\dot{\alpha}_{i}/\alpha_{i})$ is scale
independent111By a dot we denote the time-derivative of the coupling constant.
Thus, $\dot{\alpha}$ denotes the instantaneous time change of the fine-
structure constant. The capital Greek letter $\Delta$, while not used in this
paper, represents the difference in value of the coupling constant at
different times, e.g., $\Delta\alpha=\alpha(t_{2})-\alpha(t_{1})$ with
$t_{2}>t_{1}$. Finally, we will use the symbol $\delta$ to denote the time-
dependent part of the coupling constant. For example, $\delta\alpha$ denotes
the time-dependent part of the function $\alpha(t)$. Note that, sadly, authors
in this field do not often define their notations carefully which may be the
source of some confusion..

We shall now calculate the relation between the time dependence of the fine
structure constant $\alpha$ and the quantum chromodynamics scale $\Lambda_{\rm
QCD}$. Following the same arguments as those presented in CF2002 , we obtain a
relation between the instantaneous time changes of $\alpha$ and $\Lambda_{\rm
QCD}$ given by

$\displaystyle\frac{\dot{\Lambda}_{\rm QCD}}{\Lambda_{\rm
QCD}}=R\frac{\dot{\alpha}}{\alpha}(\mu=0).$ (10)

The quantity $R$ is calculated in CF2002  and found to take the value
$R=37.7\pm 2.3$. The uncertainty is due to low energy non-perturbative quantum
chromodynamics uncertainties. The quantity $R$ is a prediction of grand
unification which can be probed by low energy experiments such as clocks.

The effective Lagrangian (4) is defined at the unification scale $M_{u}$. We
can run it down to a low-energy scale $\mu$ and obtain

${\cal
L}=-\frac{1}{4}\sum_{i=1}^{3}\left(\frac{1}{g_{u}^{2}}+\sum_{r}\frac{c_{r}}{M_{\rm
P}}H_{r}\delta_{i}^{(r)}+\frac{b_{i}}{8\pi^{2}}\log\left(\frac{M_{u}}{\mu}\right)\right)F_{\mu\nu(i)}{F}^{\mu\nu}_{(i)}.$
(11)

Below the supersymmetry breaking scale $M_{\rm S}$, one has

${\cal
L}=-\frac{1}{4}\sum_{i=1}^{3}\left(\frac{1}{g_{u}^{2}}+\sum_{r}\frac{c_{r}}{M_{\rm
P}}H_{r}\delta_{i}^{(r)}+\frac{b^{\rm{S}}_{i}}{8\pi^{2}}\log\left(\frac{M_{u}}{M_{\rm
S}}\right)+\frac{b^{\rm{SM}}_{i}}{8\pi^{2}}\log\left(\frac{M_{\rm
S}}{\mu}\right)\right)F_{\mu\nu(i)}{F}^{\mu\nu}_{(i)}.$ (12)

As we are working to leading order in perturbation theory this is equivalent
to

${\cal
L}=-\frac{1}{4}\sum_{i=1}^{3}\left(1+g_{i}^{2}(\mu)\sum_{r}\frac{c_{r}}{M_{\rm
P}}H_{r}\delta_{i}^{(r)}\right)F_{\mu\nu(i)}{F}^{\mu\nu}_{(i)}.$ (13)

Below the electroweak breaking energy scale, we can reformulate the effective
Lagrangian in terms of the mass eigenstates: the photon $F_{\mu\nu}$ and the
three massive electroweak bosons $Z_{\mu\nu},W_{\mu\nu}$. In terms of the mass
eigenstates and the $\epsilon_{i}$, the effective Lagrangian reads

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left(1+e^{2}(\mu)\left(\epsilon_{1}+\epsilon_{2}\right)\right)F_{\mu\nu}{F}^{\mu\nu}$
$\displaystyle-\frac{1}{4}\left(1+e^{2}(\mu)\left(\epsilon_{1}\tan^{2}\theta_{W}(\mu)+\epsilon_{2}\cot^{2}\theta_{W}(\mu)\right)\right)Z_{\mu\nu}{Z}^{\mu\nu}$
$\displaystyle-\frac{1}{2}\left(1+g_{2}^{2}(\mu)\epsilon_{2}\right){\rm
Tr}W^{\dagger}_{\mu\nu}W^{\mu\nu}-\frac{1}{2}\left(1+g_{3}^{2}(\mu)\epsilon_{3}\right){\rm
Tr}G_{\mu\nu}{G}^{\mu\nu}.$

As expected, the photon and $Z$ boson interactions show the expected mixing
between $U(1)_{Y}$ and $SU(2)_{L}$ couplings $\epsilon_{1},\epsilon_{2}$.
Similarly, the $W$ boson and gluon terms are exclusively associated with
$\epsilon_{2}$ and $\epsilon_{3}$, respectively. Modulo the Clebsch-Gordon
coefficients $\delta_{i}^{(r)}$, and as a consequence of the embedding in a
unified theory, all modifications are proportional to the combination
$\sum_{r}(c_{r}/M_{\rm P})H_{r}$ and the coupling-constant variations are
interdependent.

From Eqs. (6),(II), we identify the fine-structure constant $\alpha$ and
strong coupling constant $\alpha_{s}$:

$\displaystyle\alpha(\mu)$ $\displaystyle=$
$\displaystyle\frac{\bar{e}^{2}(\mu)}{4\pi}=\frac{e^{2}(\mu)(1-e^{2}(\mu)(\epsilon_{1}+\epsilon_{2}))}{4\pi},$
(15) $\displaystyle\alpha_{s}(\mu)$ $\displaystyle=$
$\displaystyle\frac{\bar{g}_{3}^{2}(\mu)}{4\pi}=\frac{g_{3}^{2}(\mu)(1-g_{3}^{2}(\mu)\epsilon_{3})}{4\pi},$
(16)

which gives the instantaneous time changes

$\displaystyle\frac{\dot{\alpha}(\mu)}{\alpha(\mu)}\approx-4\pi\alpha(\mu)(\dot{\epsilon}_{1}+\dot{\epsilon}_{2}),\qquad\frac{\dot{\alpha}_{s}(\mu)}{\alpha_{s}(\mu)}\approx-4\pi\alpha_{s}(\mu)\dot{\epsilon}_{3}.$
(17)

Similarly, the instantaneous time change of $\Lambda_{\rm QCD}$ is given by

$\displaystyle\frac{\dot{\Lambda}_{\rm QCD}}{\Lambda_{\rm QCD}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\log\left(\frac{\mu^{2}}{\Lambda_{\rm
QCD}^{2}}\right)\frac{\dot{\alpha}_{s}(\mu)}{\alpha_{s}(\mu)}\approx\frac{8\pi^{2}}{b_{3}}\dot{\epsilon}_{3},$
(18)

where $b_{3}=-11+(2/3)n_{f}$ for $n_{f}$ quark flavors. We thus find a generic
prediction of grand unification between the time change of the scale of
quantum chromodynamics and that of the fine structure constant. Note that this
relation is model independent in the sense that it does not depend on the
scale at which supersymmetry is broken or the unification group. A time shift
of the scale of quantum chromodynamics implies a time shift of the nucleon
mass since its mass is proportional to that scale and determined to 90$\%$ by
gluonic dynamics. Grand unification could simply be probed if time changes of
the nucleon mass and the fine structure constant were observed. Grand
unification predicts the ratio of these time variation (see Eq. (10)) in a
model independent manner.

##  III Ultra light dark matter and constraints

Thus far, we have treated the scalar multiplets $H^{ab}_{r}$ as generic time-
dependent classical fields. Let us now consider the case where $H_{r}$ are
ultra light and represent dark matter. At energies $\mu\lesssim 1$ GeV of
atomic-scale experiments, electrons, light quarks, and their associated gauge
fields are the most relevant degrees of freedom. From this point of view, the
interesting part of the effective Lagrangian Eq. (II) are the first and last
terms parametrizing the interactions of four ultra light dark-matter
candidates to photons and gluons. Given this assumption, the functional form
$H_{r}(t)$ is well approximated by a single-mode solution to the Klein-Gordon
equation DP14 ; SF14 ; AHT15

$\displaystyle
H_{r}(t)\approx\frac{\sqrt{2\rho_{H_{r}}}}{m_{H_{r}}}\cos(m_{H_{r}}t),$ (19)

where $m_{H_{r}}$ is the mass and $\rho_{H_{r}}$ is some fraction of the dark-
matter density ($\rho_{\rm DM}\approx 0.4$ GeV/cm3). This solution generates
periodic oscillations of the fine-structure constant $\alpha$ and strong
coupling constant $\alpha_{s}$.

Searches for ultra light dark matter have been performed in a variety of
experiments SMDM . As applications, we consider three types of experiments
that together yield the most stringent constraints on the coefficients $c_{r}$
in the mass range $10^{-24}$ eV $\lesssim m_{H_{r}}\lesssim 10^{-14}$ eV.
These include optical atomic clocks, in particular those of the Physikalisch-
Technische Bundesanstalt (PTB) Filzinger23 ; pulsar timing array observations
from the NANOGrav Collaboration KMT22 ; NANOGrav23 ; and weak equivalence
principle tests by the MICROSCOPE Collaboration Berge:2017ovy ;
MICROSCOPE:2022doy .

Optical atomic clock experiments measure the ratio $f_{A}/f_{B}$ of two
transitions $A$ and $B$. The time-dependent part of this ratio is typically
expressed as a relative fraction and parametrized in terms of the relative
time-dependent contributions of the fine-structure constant $\alpha$, electron
mass to QCD scale $m_{e}/\Lambda_{\rm QCD}$, and light-quark masses to QCD
scale $m_{q}/\Lambda_{\rm QCD}$ as Flambaum04

$\displaystyle\frac{\delta(f_{A}/f_{B})}{f_{A}/f_{B}}$
$\displaystyle=(k_{A}^{\alpha}-k_{B}^{\alpha})\frac{\delta\alpha}{\alpha}+(k_{A}^{\mu}-k_{B}^{\mu})\frac{\delta(m_{e}/\Lambda_{\rm
QCD})}{m_{e}/\Lambda_{\rm
QCD}}+(k_{A}^{q}-k_{B}^{q})\frac{\delta(m_{q}/\Lambda_{\rm
QCD})}{m_{q}/\Lambda_{\rm QCD}}$
$\displaystyle=(k_{A}^{\alpha}-k_{B}^{\alpha})\frac{\delta\alpha}{\alpha}-(k_{A}^{\mu}-k_{B}^{\mu}+k_{A}^{q}-k_{B}^{q})\frac{\delta\Lambda_{\rm
QCD}}{\Lambda_{\rm QCD}},$ (20)

where the proportionality factors $k_{A}^{\alpha},k_{A}^{\mu}$, and
$k_{A}^{q}$ quantify the sensitivity of the transition $A$ to the associated
dimensionless quantity. Note that since the effective Lagrangian Eq. (II)
contains only gauge-sector modifications by assumption222Note that in
realistic models, one expects the time variation of lepton and quark masses to
be subdominent effects CF2002 ., there is no time dependence of the electron
$m_{e}$ and quark $m_{q}$ masses ($\delta m_{e}=\delta m_{q}=0$). From Eq.
(II), we extract the relative time-dependent pieces333With these definitions,
e.g., $\dot{\alpha}=d(\delta\alpha)/dt$.

$\frac{\delta\alpha}{\alpha}\approx-4\pi\alpha(\epsilon_{1}+\epsilon_{2}),\qquad\frac{\delta\alpha_{s}}{\alpha_{s}}\approx-4\pi\alpha_{s}\epsilon_{3},\qquad\frac{\delta\Lambda_{\rm
QCD}}{\Lambda_{\rm QCD}}\approx\frac{8\pi^{2}}{b_{3}}\epsilon_{3}.$ (21)

Substituting these with Eq. (19) into Eq. (III) gives

$\frac{\delta(f_{A}/f_{B})}{f_{A}/f_{B}}=\frac{1}{M_{\rm
P}}\sum_{r}(k^{r}_{A}-k^{r}_{B})c_{r}\frac{\sqrt{2\rho_{H_{r}}}}{m_{H_{r}}}\cos(m_{H_{r}}t),$
(22)

where

$\displaystyle
k^{r}_{A}-k^{r}_{B}\equiv(k_{A}^{\alpha}-k_{B}^{\alpha})(-4\pi\alpha)(\delta_{1}^{(r)}+\delta_{2}^{(r)})-(k_{A}^{\mu}-k_{B}^{\mu}+k_{A}^{q}-k_{B}^{q})(8\pi^{2}/b_{3})\delta_{3}^{(r)}.$
(23)

If present, the signal Eq. (22) would appear as a single peak, or multiple
peaks, in the Fourier amplitude spectrum at angular frequencies $\omega\approx
m_{H_{r}}$. PTB have measured the 171Yb+ electric-octupole (E3) transition
relative to electric-quadrupole (E2) transition ($f_{\rm E3}/f_{\rm E2}$) and
the E3 transition relative to the 87Sr (1S${}_{0}\leftrightarrow{}^{3}$P0)
transition ($f_{\rm E3}/f_{\rm Sr})$ over approximately 2 years Filzinger23
,AxionLimits . These optical-to-optical comparisons are primarily
sensitive444It has been suggested that optical-clock comparisons also admit
sensitivity to nuclear radius oscillations, which could be sourced by the
second term in Eq. (24) Banerjee:2023bjc . However, the associated sensitivity
factor is estimated to be suppressed relative to $k^{\alpha}$ by $\approx
10^{3}$, resulting in significantly weaker constraints on $d_{g}$. time-
dependent contributions $\delta\alpha/\alpha$ from Eq. (III). The non-
observation of significant peaks in the amplitude spectrum was translated into
constraints on $|d_{\gamma}|$ from single dilaton-like field $\phi$ model
DD2010

$\mathcal{L}_{\phi}=\frac{1}{\sqrt{2}M_{\rm
P}}\phi\left[\frac{1}{4e^{2}}d_{\gamma}F_{\mu\nu}F^{\mu\nu}-\frac{\beta_{3}}{g_{3}}d_{g}{\rm
Tr}G_{\mu\nu}G^{\mu\nu}\right],$ (24)

in the range $10^{-23}$ eV $\lesssim m_{\phi}\lesssim 10^{-17}$ eV, where
$\beta_{3}=-b_{3}g_{3}^{3}/(16\pi^{2})$ is the quantum chromodynamics beta
function. In this model,
$\delta\alpha/\alpha=d_{\gamma}\phi(t)/(\sqrt{2}M_{\rm P})$ with
$\phi(t)=(\sqrt{2\rho_{\rm DM}}/m_{\phi})\cos(m_{\phi}t)$ in analogy with Eq.
(19). Integrating out the $Z$ boson, $W$ boson, and heavy quarks from Eq.
(II), comparing at $\mu\approx 1$ GeV, and assuming only one of the four
irreducible representations $r$ is present, allows constraints on $d_{\gamma}$
and $d_{g}$ to be recast in terms of constraints on the coefficients $c_{r}$.
Comparing with the fractional shift with those obtained from the dilaton model
Eq. (24), we find that, for a given irreducible representation $r$:

$\displaystyle\sqrt{2}(-4\pi\alpha)(\delta_{1}^{(r)}+\delta_{2}^{(r)})c_{r}$
$\displaystyle\leftrightarrow d_{\gamma},$
$\displaystyle\sqrt{2}(8\pi^{2}/b_{3})\delta_{3}^{(r)}c_{r}$
$\displaystyle\leftrightarrow d_{g}.$ (25)

Under the assumption that a single irreducible representation $r$ is present
and accounts for the total dark-matter density ($\rho_{H_{r}}=\rho_{\rm DM})$,
constraints on each of the $c_{r}$ may be extracted one at a time by setting
the remaining three to zero.

Pulsar timing arrays provide sensitivity to ultra light dark matter through
timing residuals $h(t)$, which quantify the difference between the time of
arrival of a pulsar’s measured electromagnetic signal relative to a model
prediction KMT22 . Cesium atomic clocks are used as references for pulsar
time-of-arrival measurements and are sensitive to the time-dependent
contributions $\delta f_{\rm Cs}/f_{\rm Cs}$ induced by ultra light dark
matter through the time-dependent pieces of $\alpha$, $m_{e}/\Lambda_{\rm
QCD}$, and $m_{q}/\Lambda_{\rm QCD}$ from Eq. (III)555Note that here one must
also account for the spatial coherence properties of ultra light dark matter
(see KMT22 ).. A change of the reference cesium clocks’ rate would appear as a
change in the pulsar rotation frequency $f$ (i.e. $\delta f_{\rm Cs}/f_{\rm
Cs}=\delta f/f$), the latter of which is related to the timing residuals
$h(t)$ via

$h(t)=-\int\frac{\delta f}{f}dt.$ (26)

Applications of the NANOGrav Collaboration 15-year data set include
constraints on the ultra light dark matter dilaton coefficients
$d_{\gamma},d_{g}$ NANOGrav23 . The constraints on $d_{g}$ are roughly an
order of magnitude more stringent than microwave atomic-clock measurements
Hees:2016gop  for masses $10^{-24}$ eV $\lesssim m_{\phi}\lesssim 10^{-23}$
eV. This translates to a dominant sensitivity to our $c_{r}$ coefficients
through the identification Eq. (III) over this same range.

The presence of ultra light bosonic666In this case $H_{r}$ need not
necessarily be identified with dark matter. fields $H_{r}$ will also generate
a fifth force, corresponding to a Yukawa potential between two objects $A,B$
of different composition. The MICROSCOPE Collaboration have obtained the most
stringent limit on the Eötvös parameter
$\eta=2|\vec{a}_{A}-\vec{a}_{B}|/|\vec{a}_{A}+\vec{a}_{B}|\approx 10^{-15}$
using titanium and platinum alloys MICROSCOPE:2022doy . Matching $\eta$ to a
combination of Wilson coefficients of the effective photon and gluon operators
in Eq. (II) and functions of the proton and mass numbers of $A$ and $B$
Berge:2017ovy ; Brzeminski:2022sde ; Hees:2018fpg  enables constraints on
$c_{r}$.

Figure 1: Constraints on $c_{r}$. Multiplets $r=$ 1, 24, 75, 200 are
represented by green, blue, red, and purple lines, respectively. Lighter
(darker) variations of the color palette apply to NANOGrav (MICROSCOPE).

Constraints at 95% confidence level on $c_{r}$ are shown in Fig. 1. We have
assumed a renormalization scale $\mu$ corresponding to the typical scale of
atomic physics ($\alpha\approx 1/137$) and $b_{3}=-9$, the latter
corresponding to $n_{f}=3$ light-quark flavors (and subject to theory
uncertainties from non-perturbative quantum chromodynamics—see Eq. (10)). The
small differences in $c_{r}$ exclusion regions are completely controlled by
the Clebsch-Gordon coefficients $\delta_{i}^{(r)}$ listed in Table 1. As
mentioned, NANOGrav gives the most sensitivity in the lowest mass range
$10^{-24}$ eV $\lesssim m_{H_{r}}\lesssim 10^{-23}$ eV and along with PTB
exceed the MICROSCOPE constraints until around $m_{H_{r}}\approx 10^{-20}$ eV.
We note that several astrophysical and cosmological observations indicate
$m_{H_{r}}\gtrsim 10^{-21}$ eV Ferreira20  and that lighter dark-matter
candidates can only account for $\sim 10\%$ of the total dark-matter density
$\rho_{\rm DM}$, which is also largest source of theory uncertainty for the
NANOGrav and PTB bounds. As a visual preference, we show the MICROSCOPE
constraints through $m_{H_{r}}\approx 10^{-16}$ eV though they hold at the
same magnitudes until $m_{H_{r}}\approx 10^{-14}$ eV before rapidly falling
off when the wavelength of $H_{r}$ becomes comparable to the radius of the
Earth.

Grand unification can naturally accommodate multiple scalar dark matter
candidates. In $SU(5)$ grand unified theories, each of the scalar
representations $r=$ 1, 24, 75, 200 contains one neutral scalar field that
could be a good dark matter candidate. They could all contribute to the local
dark matter density. In the (rather restrictive) case of equal masses, we note
that, assuming $\sum_{r}\rho_{H_{r}}=\rho_{\rm DM}$, constraints may placed
analogously to the single-field case on the linear combination
$\sum_{r}(k^{r}_{A}-k^{r}_{B})c_{r}\sqrt{2\rho_{H_{r}}}$.

##  IV Conclusions

In this paper, we have shown that scalar multiplets in grand unified theories
can naturally lead to a time-variation of fundamental constants via dimension
five operators expected to be generated by quantum gravity. These scalar
multiplets naturally provide a mechanism for a time variation of the gauge
coupling of the unified theory. We illustrate the mechanism using
supersymmetric $SU(5)$ grand unified theory, but our conclusions would apply
to any grand unified gauge theory. We emphasize that the unification of the
gauge couplings of the standard model can be probed at very low energy with
quantum sensors and other tabletop experiments. Grand unification imposes a
relation the time variation of the fine-structure constant and that of the
scale of quantum chromodynamics or equivalently the proton mass. Our results
do not depend on the details of the unification group or on the scale at which
supersymmetry is broken.

We then propose to identify the new scalar multiplets with ultra light dark
matter. This assumption implies a specific functional dependence on time for
the scalar multiplets. We use data from PTB, NANOGrav and MICROSCOPE to set a
limit on the coefficient of the dimension five operators generated by quantum
gravity.

Acknowledgments: This work is supported in part by the Science and Technology
Facilities Council (grant number ST/T006048/1).

Data Availability Statement: This manuscript has no associated data. Data
sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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