# Space-time variation of the $s$ and $c$ quark masses

V. V. Flambaum and P. Munro-Laylim  School of Physics, University of New South
Wales, Sydney 2052, Australia

###### Abstract

Space-time variation of fundamental physical constants in expanding Universe
is predicted by a number of popular models. The masses of second generation
quarks are larger than first generation quark masses by several orders of
magnitude, therefore space-time variation in quark masses may significantly
vary between each generation. We evaluate limits on variation in the $s$ and
$c$ quark masses from Big Bang nucleosynthesis, Oklo natural nuclear reactor,
Yb+, Cs and Rb clock data. The construction of a 229Th nuclear clock is
expected to enhance these limits by several orders of magnitude. Furthermore,
constraints are obtained on an oscillating scalar or pseudoscalar cold dark
matter field, as interactions of the field with quarks produce variations in
quark masses.

##  I Introduction

The discovery of dark matter continues to elude physicists despite it
accounting for $85\%$ of the total matter density in the Universe.Among the
range of proposed models for dark matter, axions are one of the most promising
candidates. Originally introduced to preserve $CP$-symmetry in QCD Peccei and
Quinn (1977a, b), axions and axionlike particles are used to describe light
pseudoscalar fields. Light scalar particles with dilaton-like interaction with
Standard Model particles are motivated by superstring theory Taylor and
Veneziano (1988); Damour and Polyakov (1994a, b); Damour  _et al._ (2002a, b)
and chameleon models of gravity, see, e.g., Ref. Burrage and Sakstein (2018)
and referenced therein.

Interactions between axions or light scalar dark matter fields with Standard
Model particles can lead to the variation of fundamental constants in space-
time Arvanitaki  _et al._ (2015); Stadnik and Flambaum (2015, 2016); Van
Tilburg  _et al._ (2015).

There are several new opportunities to search for the variation of fundamental
constants, namely the recent launch a network of atomic clocks known as QSNET
Barontini  _et al._ (2022) and a new proposed method to use laser-
interferometric gravitational-wave detectors Grote and Stadnik (2019).
Furthermore, as was proposed in Ref. Flambaum (2006), the transition frequency
between the ground and first excited states of the 229Th nucleus is highly
sensitive to the variation of fundamental constants. A recent review Beeks
_et al._ (2021) outlined the continual advancements towards the construction
of a 229Th nuclear clock, which will hopefully allow us to obtain strong
limits on the variation of fundamental constants in the near future.

Previous works have used various phenomena to constrain variation in the fine
structure constant $\alpha$, electron mass $m_{e}$ and the light quark mass
$m_{q}=(m_{u}+m_{d})/2$ Flambaum and Shuryak (2003); Flambaum and Tedesco
(2006); Flambaum and Wiringa (2007, 2009); Stadnik and Flambaum (2015),
however there is a lack of investigation into variation in the $s$ and $c$
quark masses. The masses of the second generation of quarks, $m_{s}=93$ MeV
and $m_{c}=1.3$ GeV, are orders of magnitude bigger than the masses of first
generation, $m_{q}=3.5$ MeV. Therefore, variation of the second generation
quark masses may significantly differ from the first generation. Ref. Flambaum
and Shuryak (2003); Flambaum and Tedesco (2006) examined variation in the
strange quark mass $m_{s}$ from a variety of phenomena, but their findings are
dated and require revision. We provide updated limits on $m_{s}$ and also
present first results on the variation of the charm quark mass, $m_{c}$.

We place constraints from variation of deuteron binding energy since Big Bang
nucleosynthesis, the Oklo natural nuclear reactor, variation in the proton-
electron mass ratio, and variation in nuclear magnetic moments. We also
determine the constraints expected from a 229Th nuclear clock.

Note that units which are used for the interpretation of the measurements also
may vary and this could introduce ambiguity in the interpretation of the
results if one considers variation of the dimensionful parameters - see
discussion in Refs. Flambaum and Shuryak (2003); Flambaum and Tedesco (2006).
Nucleon masses and strong interactions depend mainly on the QCD scale
$\Lambda_{QCD}$ and quark masses. Therefore, our results may be interpreted as
the measurements of the dimensionless parameter $X_{q}=m_{q}/\Lambda_{QCD}$
which does not depend on units which one uses. It is convenient to assume that
QCD scale $\Lambda_{QCD}$ does not vary. We may say that we measure quark
masses in units of $\Lambda_{QCD}$.

The variation in quark masses can be due to quark interactions with dark
matter. In this work, we consider a non-relativistic scalar or pseudoscalar
cold dark matter field, $\phi=\phi_{0}\cos(\omega t)$, which oscillates with
frequency $\omega=m_{\phi}c^{2}/\hbar$, where $m_{\phi}$ is mass of the spin-0
dark matter particle. Quadratic-in-$\phi$ interactions between the scalar or
pseudoscalar field and Standard Model fermion fields, $f$, are expressed as

$\mathcal{L}=\pm\sum_{f}\frac{\phi^{2}}{(\Lambda^{\prime}_{f})^{2}}m_{f}\bar{f}f,$
(1)

where $m_{f}$ is the fermion mass and $\Lambda^{\prime}_{f}$ is a large energy
scale that can differ between fermions. Comparison to the Standard Model
Lagrangian $\mathcal{L}=m_{f}\bar{f}f$ indicates that the fermion masses are
altered by the dark matter field

$m_{f}\rightarrow
m_{f}\left(1\pm\frac{\phi^{2}}{(\Lambda^{\prime}_{f})^{2}}\right).$ (2)

Quadratic interactions of $\phi$ mean that there is an oscillating component
of $\phi^{2}$, $\phi_{0}^{2}\cos(2\omega t)/2$, and a non-oscillating
component, $\phi_{0}^{2}/2$. Therefore, the fundamental constants can
experience slow non-oscillating variation, with Ref. Stadnik and Flambaum
(2015) obtaining constraints on the field for interactions with photons,
electrons, light quarks, and massive vector bosons. Our limits on variation in
$m_{s}$ and $m_{c}$ are used to constrain the non-oscillating component of
dark matter field. Linear-in-$\phi$ interactions of the scalar field with
fermions can also be considered to produce only oscillating fermion mass
variation

$\mathcal{L}=-\sum_{f}\frac{\phi}{\Lambda_{f}}m_{f}\bar{f}f,$ (3)
$m_{f}\rightarrow m_{f}\left(1+\frac{\phi}{\Lambda_{f}}\right).$ (4)

We use experimental results from Rb/Cs atomic fountain clocks Hees  _et al._
(2016) to obtain limits on dark matter interactions with second generation
quarks. Note that for linear-in-$\phi$ interactions, the field can only be
scalar.

##  II Nucleon and Meson Mass Variation

To evaluate limits on the mass variation of second generation quarks, we first
need to obtain the sensitivity in nucleon mass to variation in $m_{s}$ and
$m_{c}$. In a recent review, Ref. Aoki  _et al._ (2020) analysed and averaged
a range of lattice QCD results, from which we will use the averaged sigma
terms

$\displaystyle\sigma_{s}=m_{s}\langle N|\bar{s}s|N\rangle=$ $\displaystyle 53$
$\displaystyle{\rm MeV},$ (5) $\displaystyle\sigma_{c}=m_{c}\langle
N|\bar{c}c|N\rangle=$ $\displaystyle 78$ $\displaystyle{\rm MeV},$ (6)

where the values are averaged from Ref. Dürr  _et al._ (2012, 2016); Yang  _et
al._ (2016); Junnarkar and Walker-Loud (2013); Freeman and Toussaint (2013)
for $\sigma_{s}$ and Ref. Junnarkar and Walker-Loud (2013); Bali  _et al._
(2016); Abdel-Rehim  _et al._ (2016); Gong  _et al._ (2013); Freeman and
Toussaint (2013) for $\sigma_{c}$. Therefore, the nucleon mass sensitivity to
each of the second generation quark masses from the strange and charm seas are
found to be

$\displaystyle\frac{\delta m_{N}}{m_{N}}=\frac{\sigma_{s}}{m_{N}}\frac{\delta
m_{s}}{m_{s}}=0.056\frac{\delta m_{s}}{m_{s}},$ (7) $\displaystyle\frac{\delta
m_{N}}{m_{N}}=\frac{\sigma_{c}}{m_{N}}\frac{\delta
m_{c}}{m_{c}}=0.083\frac{\delta m_{c}}{m_{c}},$ (8)

for nucleon mass $m_{N}=939$ MeV. Our limits on second generation quark mass
variation from deuteron binding energy variation and the Oklo nuclear reactor
also require us to account for variation in the strong nuclear potential. The
Walecka model Walecka (1974) is used to express the strong nuclear potential
via the exchange of $\sigma$ and $\omega$ mesons. We evaluate the
sensitivities of meson masses to quark mass variation (see Appendix A)

$\displaystyle\frac{\delta m_{\sigma}}{m_{\sigma}}$ $\displaystyle=$
$\displaystyle 0.33\frac{\delta m_{s}}{m_{s}},$ (9) $\displaystyle\frac{\delta
m_{\sigma}}{m_{\sigma}}$ $\displaystyle=$ $\displaystyle 0.10\frac{\delta
m_{c}}{m_{c}},$ (10) $\displaystyle\frac{\delta m_{\omega}}{m_{\omega}}$
$\displaystyle=$ $\displaystyle 0.045\frac{\delta m_{s}}{m_{s}},$ (11)
$\displaystyle\frac{\delta m_{\omega}}{m_{\omega}}$ $\displaystyle=$
$\displaystyle 0.067\frac{\delta m_{c}}{m_{c}}.$ (12)

##  III Variation in second generation quark masses

###  III.1 Big Bang Nucleosynthesis

In Ref. Flambaum and Shuryak (2003), variation in deuteron binding energy was
estimated using the Walecka model

$\frac{\delta Q_{d}}{Q_{d}}=-48\frac{\delta
m_{\sigma}}{m_{\sigma}}+50\frac{\delta m_{\omega}}{m_{\omega}}+6\frac{\delta
m_{N}}{m_{N}}.$ (13)

Note that the Walecka model cannot be used to correctly describe deuteron
binding energy variation, as the tensor forces (from $\pi$ and $\rho$
exchange) are required to account for all spin-dependent forces. Instead of
accounting for the tensor forces directly, the authors of Ref. Flambaum and
Shuryak (2003) modified the Walecka model by reducing $g_{v}^{2}$ by a factor
of 0.953 to obtain Eq. 13. This estimate allowed them to correctly evaluate
the deuteron binding energy. Therefore, using our results for nucleon and
meson mass variation, we find the sensitivity in deuteron binding energy due
to second generation quark mass variation

$\displaystyle\frac{\delta Q_{d}}{Q_{d}}$ $\displaystyle=$
$\displaystyle-13\frac{\delta m_{s}}{m_{s}},$ (14) $\displaystyle\frac{\delta
Q_{d}}{Q_{d}}$ $\displaystyle=$ $\displaystyle-0.95\frac{\delta
m_{c}}{m_{c}}.$ (15)

From Ref. Dmitriev  _et al._ (2004), the limit on the variation of deuteron
binding energy from Big Bang nucleosynthesis to today is $\delta
Q_{d}/Q_{d}=-0.019\pm 0.005$. Therefore, we obtain limits on the mass
variation of second generation quarks since Big Bang nucleosynthesis

$\displaystyle\left|\frac{\delta m_{s}}{m_{s}}\right|$ $\displaystyle<$
$\displaystyle 0.0018,$ (16) $\displaystyle\left|\frac{\delta
m_{c}}{m_{c}}\right|$ $\displaystyle<$ $\displaystyle 0.025.$ (17)

###  III.2 Oklo natural nuclear reactor

The Oklo natural nuclear reactor is a self-sustaining nuclear fission reactor
that has been active for around 2 billion years. At Oklo, the disappearance of
isotopes with near-zero neutron resonance energy, most notably 149Sm, can be
used to constrain the shift in the lowest resonance, $\delta E$. The strongest
limit on this energy shift is $|\delta E|<0.02$ eV Damour and Dyson (1996).
From Ref. Flambaum and Shuryak (2003), the variation of the resonance position
is given by

$\displaystyle\delta E=V_{0}\left(8.6\frac{\delta
m_{\sigma}}{m_{\sigma}}-6.6\frac{\delta m_{\omega}}{m_{\omega}}-\frac{\delta
m_{N}}{m_{N}}\right),$ (18)

from which we obtain the variation of resonance position due to $s$ and $c$
quark mass variation, with $V_{0}\approx 50$ MeV,

$\displaystyle\delta E$ $\displaystyle=$ $\displaystyle 120\;{\rm
MeV}\times\frac{\delta m_{s}}{m_{s}},$ (19) $\displaystyle\delta E$
$\displaystyle=$ $\displaystyle 17\;{\rm MeV}\times\frac{\delta
m_{c}}{m_{c}}.$ (20)

Therefore, we constrain the mass variation of second generation quarks

$\displaystyle\left|\frac{\delta m_{s}}{m_{s}}\right|$ $\displaystyle<$
$\displaystyle 1.7\times 10^{-10},$ (21) $\displaystyle\left|\frac{\delta
m_{c}}{m_{c}}\right|$ $\displaystyle<$ $\displaystyle 1.2\times 10^{-9}.$ (22)

Assuming a constant rate of variation over $1.8\times 10^{9}$ years, these
limits correspond to the constrained rate of variation

$\displaystyle\left|\frac{1}{m_{s}}\frac{dm_{s}}{dt}\right|$ $\displaystyle<$
$\displaystyle 9.4\times 10^{-20}\;{\rm yr}^{-1},$ (23)
$\displaystyle\left|\frac{1}{m_{c}}\frac{dm_{c}}{dt}\right|$ $\displaystyle<$
$\displaystyle 6.7\times 10^{-19}\;{\rm yr}^{-1}.$ (24)

###  III.3 Proton-electron mass ratio

Recent works have measured bounds on the time variation of the proton-electron
mass ratio $\mu_{pe}=m_{p}/m_{e}$ with improving accuracy Lange  _et al._
(2021); McGrew  _et al._ (2019). We assume that $m_{e}$ does not vary and
obtain limits on second generation quark mass variation, and therefore
$\delta\mu_{pe}/\mu_{pe}=\delta m_{p}/m_{p}$ with the nucleon mass in units of
electron mass. Using our previously mentioned values in Eq. 7 and 8, we get
the limits from proton-electron mass variation

$\displaystyle\frac{\delta\mu_{pe}}{\mu_{pe}}$ $\displaystyle\approx$
$\displaystyle 0.056\frac{\delta m_{s}}{m_{s}},$ (25)
$\displaystyle\frac{\delta\mu_{pe}}{\mu_{pe}}$ $\displaystyle\approx$
$\displaystyle 0.083\frac{\delta m_{c}}{m_{c}}.$ (26)

Ref. Lange  _et al._ (2021) obtained the strong limit of
$(1/\mu_{pe})(d\mu_{pe}/dt)=-8(36)\times 10^{-18}$ yr-1 from comparing
transition frequencies in optical Yb+ clocks and Cs clock. We use this result
to obtain our limits on the second generation quark mass variation

$\displaystyle\left|\frac{1}{m_{s}}\frac{dm_{s}}{dt}\right|$ $\displaystyle<$
$\displaystyle 7.9\times 10^{-16}\;{\rm yr}^{-1},$ (27)
$\displaystyle\left|\frac{1}{m_{c}}\frac{dm_{c}}{dt}\right|$ $\displaystyle<$
$\displaystyle 5.3\times 10^{-16}\;{\rm yr}^{-1}.$ (28)

###  III.4 Nuclear magnetic moments

It was first noted in Ref. Karshenboim (2000) that the ratios of hyperfine
structure intervals between different atoms are sensitive to the variation of
nuclear magnetic moments. In Ref. Flambaum  _et al._ (2004), chiral
perturbation theory was used to calculate nucleon magnetic moment variation,
$\mu_{p}$ and $\mu_{n}$ for protons and neutrons respectively, from the $s$
quark. These results were then implemented in Ref. Flambaum and Tedesco (2006)
to calculate variation of the dimensionless nuclear magnetic moment (in units
of nuclear magneton), $\mu$, for a variety of nuclei of the form

$\frac{\delta\mu}{\mu}=\kappa_{s}\frac{\delta m_{s}}{m_{s}},$ (29)

where values for $\kappa_{s}$ are nucleus-dependent and are contained in Ref.
Flambaum and Tedesco (2006). The $c$ quark is too heavy for chiral
perturbation theory, and the corresponding contribution to variation in
nuclear magnetic moments is likely small. Let us define the parameter

$V\equiv\frac{A}{E}={\rm
const}\times[\alpha^{2}F_{\mathrm{r}el}(Z\alpha)]\left(\mu\frac{m_{e}}{m_{p}}\right),$
(30)

where $A$ is the hyperfine structure constant and $E=m_{e}e^{4}/\hbar^{2}$ is
the atomic unit of energy. The first set of brackets relate the dependence on
$\alpha$ and the relativistic correction factor (Casimir factor),
$F_{\mathrm{r}el}$. The last set of brackets contain the dimensionless nuclear
magnetic moment, $\mu$ (nuclear magnetic moment $M=\mu(e\hbar/2m_{p}c)$), and
the proton and electron masses, $m_{p}$ and $m_{e}$. Assuming that only the
mass of second generation quarks varies, only $\mu$ and $m_{p}$ contribute to
the variation of $V$

$\frac{\delta V}{V}=\frac{\delta\mu}{\mu}-\frac{\delta m_{p}}{m_{p}}.$ (31)

Therefore, we consider the ratio of hyperfine structure constants,
$X(a_{1}/a_{2})=V(a_{1})/V(a_{2})$, between two atoms, $a_{1}$ and $a_{2}$.
The variation in this ratio from variation in $m_{s}$ is then expressed as

$\displaystyle\frac{\delta
X(a_{1}/a_{2})}{X(a_{1}/a_{2})}=(\kappa_{s,a_{1}}-\kappa_{s,a_{2}})\frac{\delta
m_{s}}{m_{s}}.$ (32)

Limits on slow-drift variation between 87Rb and 133Cs hyperfine transitions
from the dual atomic fountain clock FO2 at LNE-SYRTE in Ref. Abgrall  _et al._
(2015) are $\left(1/X\left({\rm Rb/Cs}\right)\right)\left(dX\left({\rm
Rb/Cs}\right)/dt\right)<-11.6(6.1)\times 10^{-17}$ yr-1 from numerous
measurements spanning over a decade. Therefore, we obtain the limit on
variation in $s$ quark mass

$\left|\frac{1}{m_{s}}\frac{dm_{s}}{dt}\right|<9.8\times 10^{-15}\;{\rm
yr}^{-1},$ (33)

where we have used $\kappa_{s,\rm Rb}=-0.010$ and $\kappa_{s,\rm Cs}=0.008$
Flambaum and Tedesco (2006).

###  III.5 Thorium nuclear clock

Ref. Flambaum (2006) noted that a 229Th nuclear clock transition between the
ground and first excited states will be highly sensitive to variations in
quark masses. Later work was done to calculate the difference in Coulomb and
kinetic energies between the $3/2^{+}$ and $5/2^{+}$ states using Hartree-Fock
(HF) and Hartree-Fock-Bogoliubov (HFB) calculations Litvinova  _et al._
(2009). The difference in Coulomb energies were used to obtain the transition
frequency sensitivity to variation in $\alpha$, while the difference in
kinetic energies $T=p^{2}/2m$ can be used to obtain sensitivity to variation
in nucleon masses. For a transition energy, $\omega=E_{1}-E_{0}$ the
sensitivity to variation in nucleon mass may be estimated as

$\displaystyle\delta\omega$ $\displaystyle=$ $\displaystyle-(\Delta
T_{p}+\Delta T_{n})\frac{\delta m_{N}}{m_{N}},$ (34)

We assume $m_{p}=m_{n}=m_{N}$. In Ref. Litvinova  _et al._ (2009), values for
$\Delta T_{p}$ and $\Delta T_{n}$ were obtained using HF and HFB calculations.
The HFB results for the energy functional SIII Beiner  _et al._ (1975) are
used due to their similarity to the experimental energy of the 229Th 5/2+
ground state, as well as close similarity to semi-empirical estimates on
Coulomb, kinetic and strong energy shifts from Ref. Fadeev  _et al._ (2022).
We obtain

$\delta\omega=-65\;{\rm keV}\times\frac{\delta m_{N}}{m_{N}}.$ (35)

As was shown in Ref. Campbell  _et al._ (2012), a 229Th nuclear clock can
reach precision $\delta\omega/\omega\sim 10^{-19}$. For the transition
frequency between the 3/2+ and 5/2+ states, $\omega\approx 8$ eV Sikorsky  _et
al._ (2020), we expect limits on variation in the transition frequency from
229Th clocks to reach $\delta\omega\approx 8\times 10^{-19}$ eV yr-1. For this
variation be due from variation in the second generation quark masses, we
obtain the expected limits attainable by a 229Th clock

$\displaystyle\left|\frac{1}{m_{s}}\frac{dm_{s}}{dt}\right|$
$\displaystyle\lesssim$ $\displaystyle 2\times 10^{-22}\;{\rm yr}^{-1},$ (36)
$\displaystyle\left|\frac{1}{m_{c}}\frac{dm_{c}}{dt}\right|$
$\displaystyle\lesssim$ $\displaystyle 1\times 10^{-22}\;{\rm yr}^{-1}.$ (37)

These expected constraints are the strongest out of all phenomena we have
investigated; the construction of a 229Th clock is highly anticipated for
investigations into the variation of fundamental constants.

##  IV Limits on scalar dark matter

###  IV.1 Limits from non-oscillating contribution

Figure 1:  From top to bottom: Limits on quadratic interaction between scalar
and pseudoscalar dark matter field $\phi$ and $s$ and $c$ quarks. Shaded
region represents excluded values. Blue region corresponds to limits from
variation in deuteron binding energy since Big Bang nucleosynthesis Dmitriev
_et al._ (2004). Red region corresponds to limits based on measurement of
oscillating effect in Ref. Hees  _et al._ (2016).

We first consider quadratic interactions of an oscillating scalar or
pseudoscalar dark matter field, $\phi=\phi_{0}\cos(\omega t)$, with $s$ and
$c$ quarks. Using Eq. 2, the fractional variation in fermion mass from the
field

$\bigg{|}\frac{\delta
m_{f}}{m_{f}}\bigg{|}=\frac{\phi^{2}}{(\Lambda^{\prime}_{f})^{2}}.$ (38)

This variation has a non-oscillating component,
$\phi_{0}^{2}/2(\Lambda^{\prime}_{f})^{2}$, which induces non-oscillating
variation of fundamental constants from changes in the dark matter density by
an oscillating field $\rho=m_{\phi}^{2}\langle\phi^{2}\rangle$, where
$\rho_{DM,0}=1.3\times 10^{-6}\;{\rm GeV\;cm}^{-3}$ is average dark matter
density Particle Data Group  _et al._ (2022). Note that there is a lower limit
$m_{\phi}>10^{-22}$ eV on the field to allow correct large scale structure
formation in galaxies Bozek  _et al._ (2015); Schive  _et al._ (2016);
Armengaud  _et al._ (2017); the field can only contribute to a fraction of the
dark matter density below this limit.

We consider limits on $\Lambda^{\prime}_{f}$ and $m_{\phi}$ for two cases:
$m_{\phi}>H$ and $m_{\phi}<H$, where $H$ is the Hubble parameter during Big
Bang nucleosynthesis. In the former case, $\phi$ remains an oscillating field,
and in the latter case, the field freezes due to Hubble friction, becoming a
non-oscillating constant field. For an early radiation-dominated universe, the
Hubble parameter scales as $H=1/2t$. Therefore, at the time of Big Bang
nucleosynthesis, we have the cross-over value $m_{\phi}\sim 10^{-16}$ eV.

Using Eq. 38 and our limits on variation in $m_{s}$ and $m_{c}$ from Eq. 16
and 17, we evaluate constraints for $\Lambda^{\prime}_{f}$ and $m_{\phi}$ (see
Appendix for details) for $m_{\phi}>H$

$\displaystyle(\Lambda^{\prime}_{s})^{2}>\frac{4.3\times 10^{17}\;{\rm
eV}^{4}}{m_{\phi}^{2}},$ (39)
$\displaystyle(\Lambda^{\prime}_{c})^{2}>\frac{3.1\times 10^{16}\;{\rm
eV}^{4}}{m_{\phi}^{2}},$ (40)

and for $m_{\phi}<H$

$\displaystyle(\Lambda^{\prime}_{s})^{2}>\frac{2.2\times 10^{44}\;{\rm
eV}^{5/2}}{m_{\phi}^{1/2}},$ (41)
$\displaystyle(\Lambda^{\prime}_{c})^{2}>\frac{1.6\times 10^{43}\;{\rm
eV}^{5/2}}{m_{\phi}^{1/2}}.$ (42)

Our results are presented in Fig. 1, where the change in gradient illustrates
the cross-over value between $m_{\phi}>H$ and $m_{\phi}<H$.

Figure 2:  From top to bottom: Limits on linear interaction between scalar
dark matter field $\phi$ and $s$ and $c$ quarks. Shaded region represents
excluded values. Blue region corresponds to limits from variation in deuteron
binding energy since Big Bang nucleosynthesis Dmitriev  _et al._ (2004). Red
region corresponds to limits based on measurement of oscillating effect in
Ref. Hees  _et al._ (2016).

We also consider linear-in-$\phi$ interactions between an oscillating scalar
dark matter field with second generation quarks. From Eq. 4, the fractional
variation in mass oscillates with the field

$\bigg{|}\frac{\delta m_{f}}{m_{f}}\bigg{|}=\frac{\phi}{\Lambda_{f}}.$ (43)

However, the field freezes when $m_{\phi}<H$, inducing slow non-oscillating
variation in fundamental constants. Therefore, similarly to the quadratic
interaction, we obtain the constraints, also presented in Fig. 2,

$\displaystyle\Lambda_{s}>\frac{3.4\times 10^{23}\;{\rm
eV}^{5/4}}{m_{\phi}^{1/4}},$ (44) $\displaystyle\Lambda_{c}>\frac{2.5\times
10^{22}\;{\rm eV}^{5/4}}{m_{\phi}^{1/4}}.$ (45)

###  IV.2 Limits from oscillating contribution

In Ref. Hees  _et al._ (2016), the dual rubidium and cesium atomic fountain
clock FO2 at LNE-SYRTE was used to search for oscillations in the ratio of
Rb/Cs hyperfine transition frequencies. As previously demonstrated in Eq. 32,
the ratio of hyperfine transition frequencies is sensitive to variations in
$m_{s}$. Since the oscillating mass variation can originate from interactions
with $\phi$, oscillations in this ratio can be used to constrain the
interactions.

The measurements for limits on the amplitude of oscillations from Ref. Hees
_et al._ (2016) are used to find limits for the linear and quadratic
interactions. Our constraints are presented in Fig. 2 Note that we use
$\phi_{0}=\sqrt{\rho_{DM}}/m_{\phi}$ for the local dark matter density
$\rho_{DM}=0.3$ GeV cm-3.

##  V Conclusion

In this work, we examined variation in strange and charm quark masses. We
obtained our strongest limits on the rate of mass variation,
$(1/m)(dm/dt)\lesssim 10^{-20}$ yr-1, using measurements from the Oklo natural
nuclear reactor in Ref. Damour and Dyson (1996). Assuming an expected accuracy
of $\delta\omega\sim 10^{-19}$ Campbell  _et al._ (2012), we show that a 229Th
nuclear clock is expected to provide the best limits of $(1/m)(dm/dt)\lesssim
10^{-22}$ yr-1.

An oscillating scalar or pseudoscalar cold dark matter field that interacts
with $s$ and $c$ quarks can produce variations in quark masses. We place
limits in Fig. 1 on previously unconstrained quadratic-in-$\phi$ interaction
parameters, $\Lambda^{\prime}_{s}$ and $\Lambda^{\prime}_{c}$, and mass of a
spin-0 scalar or pseudoscalar dark matter particle, $m_{\phi}$. Linear-
in-$\phi$ interactions between the dark matter field and second generation
quarks are also examined, where our limits on $\Lambda_{s}$, $\Lambda_{c}$ and
$m_{\phi}$ are presented in Fig. 2.

## Acknowledgements

This work was supported by the Australian Research Council Grants No.
DP190100974 and DP200100150.

##  Appendix A Meson Mass Variation

We use the Walecka model Walecka (1974) to express the strong nuclear
potential via the exchange of $\sigma$ and $\omega$ mesons

$V=-\frac{g_{s}^{2}}{4\pi}\frac{e^{-m_{\sigma}r}}{r}+\frac{g_{v}^{2}}{4\pi}\frac{e^{-m_{\omega}r}}{r}.$
(46)

For masses $m_{\sigma}\approx 500$ MeV and $m_{\omega}\approx 780$ MeV, the
coupling strengths are $g_{s}^{2}\approx 100$ and $g_{v}^{2}\approx 190$ Serot
and Walecka (1986). In the simple constituent quark picture, we estimate meson
mass sensitivity to variation in quark mass from the quark sea to be two-
thirds of the nucleon sensitivity. This leads to our results

$\displaystyle\frac{\delta m_{\sigma}}{m_{\sigma}}$ $\displaystyle=$
$\displaystyle 0.071\frac{\delta m_{s}}{m_{s}},$ (47)
$\displaystyle\frac{\delta m_{\sigma}}{m_{\sigma}}$ $\displaystyle=$
$\displaystyle 0.10\frac{\delta m_{c}}{m_{c}},$ (48)
$\displaystyle\frac{\delta m_{\omega}}{m_{\omega}}$ $\displaystyle=$
$\displaystyle 0.045\frac{\delta m_{s}}{m_{s}},$ (49)
$\displaystyle\frac{\delta m_{\omega}}{m_{\omega}}$ $\displaystyle=$
$\displaystyle 0.067\frac{\delta m_{c}}{m_{c}},$ (50)

where we use the masses $m_{\sigma}\approx 500$ MeV and $m_{\omega}\approx
780$ MeV Particle Data Group  _et al._ (2022).

For the $\sigma$ meson, it was noted in Ref. Flambaum and Shuryak (2003) that
there are additional contributions to the variation from the $s$ quark. The
$\sigma$ meson can be approximated as the $SU(3)$ singlet state
$\sigma=(\bar{u}u+\bar{d}d+\bar{s}s)/\sqrt{3}$, leading to the valence
contribution

$\frac{\delta
m_{\sigma}}{m_{\sigma}}=\frac{m_{s}\langle\sigma|\bar{s}s|\sigma\rangle}{m_{\sigma}}\frac{\delta
m_{s}}{m_{s}}=0.12\frac{\delta m_{s}}{m_{s}}.$ (51)

Furthermore, there are additional contributions from mixing with virtual
$\bar{K}K$ and $\eta\eta$ pairs (see Ref. Flambaum and Shuryak (2003) for
details)

$\frac{\delta m_{\sigma}}{m_{\sigma}}=0.14\frac{\delta m_{s}}{m_{s}}.$ (52)

Therefore, the total sensitivity in the $\sigma$ meson mass from the $s$ quark
mass is given by the sum of all contributions

$\frac{\delta m_{\sigma}}{m_{\sigma}}=0.33\frac{\delta m_{s}}{m_{s}}.$ (53)

##  Appendix B Big Bang Nucleosynthesis Limits

We first consider $m_{\phi}>H$ and use the relation for the dark matter
density

$\rho(z)=\rho_{DM,0}(1+z)^{3},$ (54)

where $\rho_{DM,0}=1.3\times 10^{-6}\;{\rm GeV\;cm}^{-3}$ is average dark
matter density Particle Data Group  _et al._ (2022) and $z$ is the redshift
parameter. The redshift parameter for deuterium formation during Big Bang
nucleosynthesis is $z=4.3\times 10^{8}$, therefore the variation in the quark
mass is

$\bigg{|}\frac{\delta m_{f}}{m_{f}}\bigg{|}=\frac{7.8\times 10^{14}\;{\rm
eV}^{4}}{m_{\phi}^{2}(\Lambda^{\prime}_{f})^{2}}.$ (55)

Using our limits on variation in $m_{s}$ and $m_{c}$ from variation in
deuteron binding energy since Bang Bang nucleosynthesis, the resulting
constraints are

$\displaystyle(\Lambda^{\prime}_{s})^{2}>\frac{4.3\times 10^{17}\;{\rm
eV}^{4}}{m_{\phi}^{2}},$ (56)
$\displaystyle(\Lambda^{\prime}_{c})^{2}>\frac{3.1\times 10^{16}\;{\rm
eV}^{4}}{m_{\phi}^{2}}.$ (57)

We now consider the case where $m_{\phi}<H$. As $z$ is very large, we assume a
radiation-dominated universe and $H=H_{0}\Omega_{r,0}^{1/2}(1+z)^{2}$ for
$H_{0}=67$ km s-1 Mpc-1 and radiation density $\Omega_{r,0}=9.26\times
10^{-5}$ Particle Data Group  _et al._ (2022). Therefore, we use Eq. 54 to get

$\displaystyle\rho=\frac{\rho_{DM,0}}{\Omega_{r,0}^{3/4}}\bigg{(}\frac{m_{\phi}}{H_{0}}\bigg{)}^{3/2}.$
(58)

The energy density for a non-oscillating field is
$\rho=m_{\phi}^{2}\langle\phi^{2}\rangle/2$, thus we obtain the variation in
quark mass

$\displaystyle\bigg{|}\frac{\delta m_{f}}{m_{f}}\bigg{|}=\frac{3.9\times
10^{41}\;{\rm eV}^{5/2}}{m_{\phi}^{1/2}(\Lambda^{\prime}_{f})^{2}}.$ (59)

Using our limits on variation in $m_{s}$ and $m_{c}$, we find the constraints

$\displaystyle(\Lambda^{\prime}_{s})^{2}>\frac{2.2\times 10^{44}\;{\rm
eV}^{5/2}}{m_{\phi}^{1/2}},$ (60)
$\displaystyle(\Lambda^{\prime}_{c})^{2}>\frac{1.6\times 10^{43}\;{\rm
eV}^{5/2}}{m_{\phi}^{1/2}}.$ (61)

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