# Oscillations of atomic energy levels induced by QCD axion dark matter

Hyungjin Kim<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Gilad Perez<EMAIL_ADDRESS>Department of Particle Physics and Astrophysics, Weizmann Institute of
Science, Rehovot, Israel 7610001

###### Abstract

Axion-gluon interaction induces quadratic couplings between the axion and the
matter fields. We find that, if the axion is an ultralight dark matter field,
it induces small oscillations of the mass of the hadrons as well as other
nuclear quantities. As a result, atomic energy levels oscillate. We use
currently available atomic spectroscopy data to constrain such axion-gluon
coupling. We also project the sensitivities of future experiments, such as
ones using molecular and nuclear clock transitions. We show that current and
near-future experiments constrain a finely-tuned parameter space of axion
models. These can compete or dominate the already-existing constraints from
oscillating neutron electric dipole moment and supernova bound, in addition to
those expected from near future magnetometer-based experiments.

††preprint: DESY-22-088

We consider axion models, consisting of a pseudo-scalar field $a$ with the
following coupling to the gluon field strength,

$\displaystyle{\cal
L}=\frac{g_{s}^{2}}{32\pi^{2}}\frac{a}{f}G_{\mu\nu}^{a}\widetilde{G}^{a\mu\nu},$
(1)

where $f$ is an axion decay constant, $g_{s}$ is a strong coupling and
$\widetilde{G}^{a\mu\nu}$ is the dual gluon field strength. Below the QCD
scale, the above axion-gluon interaction induces axion coupling to the
hadronic states. The pion mass depends on the axion field as

$m_{\pi}^{2}(\theta)=B\sqrt{m_{u}^{2}+m_{d}^{2}+2m_{u}m_{d}\cos\theta}\,.$

Here $\theta=a/f$ and $B=-\langle\bar{q}q\rangle_{0}/f_{\pi}^{2}$ with a pion
decay constant $f_{\pi}\simeq 93\,{\rm MeV}$. The resulting axion potential
can be described by $V(\theta)=-m_{\pi}^{2}(\theta)f_{\pi}^{2}$ to leading
order [1]. Due to the $\theta$-dependent potential, the axion relaxes to the
CP conserving vacuum, thereby solving the strong CP problem dynamically [2, 3,
4, 5, 6, 7, 8, 9].

Axion oscillation around its minimum may comprise dark matter (DM) in the
present universe [10, 11, 12]. If so, the pion mass develops a subdominant
oscillatory component, given by

$\displaystyle\frac{\delta
m_{\pi}^{2}}{m_{\pi}^{2}}=-\frac{m_{u}m_{d}}{2(m_{u}+m_{d})^{2}}\theta^{2}\,.$
(2)

Other nuclear quantities such as hadron masses and magnetic moments consist of
similar oscillating contributions, all induced by the effective quadratic
coupling between the axion and the matter fields. It results in a
corresponding time-variation of the atomic energy levels, which can be probed
by monitoring transition frequencies of stable frequency standards. This
method was suggested by Arvanitaki et al. [13] for dilaton/scalar DM searches
(or a relaxion DM [14]), where the DM field naturally couples to the field
strength of the strong and electromagnetic interactions as well as fermion
masses (see also [15, 16]). Various experimental techniques have been used to
search for such scalar-SM interactions [17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]. See Refs. [30, 31] for recent reviews.

The goal of this work is to assess the possibility of whether the axion-gluon
coupling can be probed by the same method, i.e. by monitoring atomic energy
levels of stable frequency standards. We claim that the same principle can be
applied to probe the coupling (1). We show the current constraints and
projections of future experiments as well as other constraints in Figure 1. We
explain the main idea below.

Figure 1: Constraints and future projections on the axion-gluon coupling are
summarized as follows: Rb/Cs clock comparison (blue) [19], H/Si comparison
(red) [24], Iodine molecular spectroscopy (brown) [28], GEO 600 gravitational
wave detector (orange) [22], 229Th nuclear isomer transition (red dashed) [13,
32], and strontium monohydroxide (green dashed) [33]. The gray dotted line is
$f=M_{\rm pl}$. The diagonal grey line is allowed parameter space for the QCD
axion, $m^{2}f^{2}\sim m_{\pi}^{2}f_{\pi}^{2}$. Other bounds, such as
oscillating neutron EDM (purple) [34], supernova 1987A [35] (light cyan), co-
magnetometer and NASDUCK [36, 37] (gray), and axion superradiance [38] (cyan),
are also included for the comparison. Projections of axion-nucleon interaction
searches, such as CASPEr-electric (blue dashed) [39] and NASDUCK (gray dashed)
[37], are also included. Spectroscopy bounds above the green solid line must
be taken carefully as the axion could develop a static profile around the
earth [40]. If such a static profile exists, it affects the propagation of DM
axion, but this parameter space is already excluded by static neutron EDM
experiments. See the main text for details.

For the purpose of demonstration, we consider the ground state hyperfine
transition in the hydrogen atom. The hyperfine structure arises due to the
interaction between the electron magnetic moment and the magnetic field
generated by the proton magnetic moment. The transition frequency of the
ground state hydrogen hyperfine structure is

$f_{H}=\frac{2}{3\pi}\frac{g_{p}m_{e}^{2}\alpha^{4}}{m_{p}}\simeq 1420\,{\rm
MHz},$

where $g_{p}=5.586$ is the proton $g$-factor. In the presence of axion dark
matter and the axion-gluon coupling, the proton $g$-factor and proton mass
develop a small oscillating component, and so does transition frequency
$f_{H}$. The fractional variation of hyperfine transition frequency can be
written as

$\displaystyle\frac{\delta f_{H}}{f_{H}}$ $\displaystyle=$
$\displaystyle\frac{\delta g_{p}}{g_{p}}-\frac{\delta m_{p}}{m_{p}}$ (3)
$\displaystyle=$ $\displaystyle\left[\frac{\partial\ln g_{p}}{\partial\ln
m_{\pi}^{2}}-\frac{\partial\ln m_{p}}{\partial\ln
m_{\pi}^{2}}\right]\frac{\delta m_{\pi}^{2}}{m_{\pi}^{2}}$
$\displaystyle\simeq$ $\displaystyle
10^{-15}\times\frac{\cos(2mt)}{m_{15}^{2}f_{10}^{2}}$

where we have defined $m_{15}=m/10^{-15}\,{\rm eV}$ and
$f_{10}=f/10^{10}\,{\rm GeV}$, and used $\partial\ln g_{p}/\partial\ln
m_{\pi}^{2}\simeq-0.17$ and $\partial\ln m_{p}/\partial\ln m_{\pi}^{2}\simeq
0.06$. For now, we take $m$ and $f$ as independent parameters to investigate
the reach of spectroscopy experiments for axion-gluon coupling search. Axion
DM background does not change fine structure constant and electron mass to the
leading order, so the variation of those quantities is ignored. The dependence
of $g_{p}$ and $m_{p}$ on the pion mass is computed by using chiral
perturbation theory at the chiral order ${\cal O}(p^{3})$ and compared with
lattice computations. See Appendix A for details. We have used
$\theta^{2}(t)=(\rho_{\rm DM}/m^{2}f^{2})[1+\cos(2mt)]$ with $\rho_{\rm
DM}\simeq 0.4\,{\rm GeV}/{\rm cm}^{3}$. A constant offset is ignored as it is
unobservable. Equation (3) suggests that the axion-gluon coupling strength
might be probed by looking for a harmonic signal in $\delta f_{H}/f_{H}$ at
the frequency $\omega=2m$.

The above discussion is more than an academic exercise. A recent experiment
performed by Kennedy et al [24] monitored hydrogen maser frequency ($f_{H}$)
together with silicon optical cavity resonance frequency ($f_{\rm Si}$) to
probe scalar DM interactions to electromagnetic field strength and electron
mass. Since the silicon optical cavity resonance frequency has a rather weak
dependence on proton mass, the fractional variation of frequencies is
dominated by that of hydrogen maser,

$\frac{\delta(f_{H}/f_{\rm Si})}{(f_{H}/f_{\rm Si})}\simeq\frac{\delta
f_{H}}{f_{H}}.$

Claimed short-term stability of transition frequency is $\sim 3\times
10^{-13}/\sqrt{\rm Hz}$. Using Eq. (3) and 33 days of experimental results
obtained in Ref. [24], we place a constraint on axion-gluon coupling, shown as
a red line in Figure 1.

Hydrogen maser is one example of many frequency standards based on hyperfine
structure. An earlier attempt to probe scalar DM based on hyperfine
transitions was made by Hees et al [19], where they used measurement of
rubidium (87Rb) and cesium (133Cs) hyperfine transitions. For the hyperfine
structure of heavier atoms, the parametric dependence of transition frequency
is similar,

$f\propto gm_{e}^{2}\alpha^{4}/m_{p},$

but the $g$-factor is replaced by that of the nucleus. The nuclear $g$-factor
can be written as a function of nucleon $g$-factor and the spin expectation
value of valence and core nucleons. Using the result of Ref. [41] together
with the nucleon $g$-factor computed in the chiral perturbation theory, we
find

$\displaystyle\frac{\partial\ln g}{\partial\ln
m_{\pi}^{2}}=\begin{cases}-0.024&\textrm{${}^{87}$Rb},\\\
+0.011&\textrm{${}^{133}$Cs},\end{cases}$ (4)

See Appendix A for details. The fractional frequency variation is therefore

$\displaystyle\frac{\delta(f_{A}/f_{B})}{(f_{A}/f_{B})}$ $\displaystyle\simeq$
$\displaystyle-0.04\frac{\delta
m_{\pi}^{2}}{m_{\pi}^{2}}\simeq-10^{-16}\frac{\cos(2mt)}{m_{15}^{2}f_{10}^{2}}$
(5)

where $A=\,^{87}{\rm Rb}$ and $B=\,^{133}{\rm Cs}$. Using the experimental
result of Rb/Cs fountain clock [19], we obtain a constraint on axion-gluon
coupling constant, which is shown as blue line in Figure 1. It is similar to
the constraint from the H/Si comparison test, but Rb/Cs constraint extends to
a much lower mass range due to its long experimental time scale.

We have only considered hyperfine transitions so far. In principle, any stable
frequency standards can be used for axion DM search as long as the transition
frequency depends on $g$-factor and/or nucleon mass. Another example is a
vibrational molecular excitation. Since the vibrational energy level depends
on $f_{\rm vib}\propto m_{p}^{-1/2}$, we find

$\displaystyle\frac{\delta f_{\rm vib}}{f_{\rm vib}}=-\frac{1}{2}\frac{\delta
m_{p}}{m_{p}}\simeq-10^{-16}\times\frac{\cos(2mt)}{m_{15}^{2}f_{10}^{2}}.$ (6)

A recent experiment performed by Oswald et al [28] used molecular transitions
in molecular iodine ($I_{2}$) to probe the variation of fundamental constants.
We use their result to place a constraint on axion-gluon coupling, which is
shown as brown in the summary figure. The constraint is relevant for the
relatively high mass end of the shown parameter space.

Better sensitivity to the variation of nucleon mass can be achieved by
considering nearly degenerate vibrational energy levels in polyatomic
molecules. It was shown in Ref. [33] that a transition between nearly
degenerate rovibrational energy levels in strontium monohydroxide (SrOH) could
yield $\delta f/f=K\delta m_{p}/m_{p}$ with $K\sim{\cal O}(10^{2})$. The
projected sensitivity from SrOH is presented as a green dashed line in Figure
1.

As a final example, we consider a nuclear transition. Refs. [42, 43] suggested
that an excitation in Thorium could be used as a frequency standard with
extraordinary accuracy due to its anomalously low excitation energy and long
lifetime. The latest measurement of such nuclear clock transition frequency is
$\omega=8.3\,{\rm eV}$ [44]. This low-lying excitation is due to the
cancellation of binding energy from strong and electromagnetic interaction. It
suggests that even a small variation either in nuclear binding energy or
electromagnetic binding energy will result in rather a large variation in
transition frequency. The variation of transition frequency as a function of
hadron mass is estimated in Refs. [45, 46]. Using their result, one finds

$\displaystyle\frac{\delta f_{\rm Th}}{f_{\rm Th}}\simeq(2\times
10^{5})\frac{\delta m_{\pi}^{2}}{m_{\pi}^{2}}.$ (7)

This large sensitivity coefficient is due to the cancellation between nuclear
and electromagnetic binding energy. Assuming that noise is dominated by
quantum projection noise of nuclear clock and that $t_{\rm int}=10^{6}\,{\rm
sec}$ of an integration time [32], we obtain the reach of nuclear clock
transition for axion-gluon coupling in Figure 1 (red dashed).

Let us now discuss limitations of the analysis as well as other existing
constraints for axion-gluon coupling in a similar mass range.

We assume that the axion comprises entire DM and that its field value on the
surface of the earth is given by the local DM density, i.e.
$\theta(t)=(\sqrt{2\rho_{\rm DM}}/mf)\cos(mt)$. The gravitational potential of
the sun might modify the field value on the surface of the earth but such
effect is generally an order of $(v_{\rm esc}/v_{\rm dm})^{2}\sim 10^{-2}$ so
that it can be ignored at the level of precision of the current analysis [47].

A more concerning effect is the finite density effect due to the sun or the
earth. As is already shown in Figure 1, current constraints and future
projections are above the QCD axion line. This range of parameter space is
fine-tuned parameter space for minimal QCD axion models. For this parameter
space, it was shown by Hook and Huang [40] that the axion potential inside the
sun or the earth could have a minimum at $\theta\neq 0$, developing static
axion profile $\theta_{s}(r)\propto e^{-m_{a}r}/r$ (see also [48, 49, 50]).
The green line in the figure corresponds to the parameter space where the
earth sources the static axion profile. Such profile acts as an attractive
potential for dark matter axion field, and hence the field value on the
surface of the earth could be enhanced compared to its local value. While we
acknowledge that this effect is likely to happen for a large part of the
parameter space shown in the figure, we do not include this effect in our
analysis since such parameter space is already constrained by experimental
limits on neutron electric dipole moment.

We have argued that spectroscopy searches can constrain or probe the parameter
space well above the QCD axion line, i.e. fine-tuned parameter space. It is
worth noting that such parameter space can be achieved in alternative QCD
axion models in a natural way. Using a large discrete symmetry and $N$-copy of
the standard model, it was shown that the parameter space above the QCD axion
line can be obtained in a technically natural way [51, 52, 53].

Other atomic spectroscopic and interferometric experiments can also be used
for searches of axion-gluon interaction. Examples include frequency comparison
between mechanical oscillators [25], unequal arm-length interferometry [23],
and interferometric gravitational wave detectors [22, 26]. All of these
experiments depend on the change of nucleon masses to some degree through the
frequency of mechanical oscillator, the refractive index of fiber that is used
in the interferometer, and the differential acceleration of test masses. As an
example, we show the constraint from GEO 600 gravitational wave detector
(orange) in the figure.

We do not consider frequency standards based on the electronic transition in
this work. This is because electronic transition frequencies scale as
$f\propto M\alpha^{\xi}$ with the reduced mass $M=m_{e}M_{N}/(m_{e}+M_{N})$ so
that the effect of axion DM in electronic transition is always suppressed by
$\propto m_{e}/M_{N}$ with nucleus mass $M_{N}$. It would still be interesting
as a future direction to investigate the reach of optical atomic clocks for
axion-gluon coupling search since uncertainties in optical clocks are much
smaller than in clocks based on hyperfine structures.

We compare spectroscopic experiments with other axion searches for axion-
nucleon interactions. The axion-gluon coupling (1) inevitably induces nucleon
electric dipole moment (EDM), $d_{n}\simeq 2.4\times
10^{-16}\bar{\theta}\,{\rm e\cdot cm}$ [54]. The axion DM background leads to
oscillating neutron EDM. By measuring spin precession frequencies of neutron
and mercury, Abel et al placed a strong constraint on axion-neutron coupling
below $m\lesssim 10^{-17}\,{\rm eV}$ [34], which is shown as the purple region
in the figure. A similar constraint was obtained from the EDM measurement of
HfF+ [55].

Magnetometers provide another way to measure axion-nucleon coupling. The same
axion-gluon coupling (1) induces axial vector coupling of axion to nucleon,
${\cal L}\supset
C_{\psi}(\partial_{\mu}a/2f)\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$ with
$C_{p}=-0.47$ and $C_{n}=0.02$ [56]. In the nonrelativistic limit, the way
that the gradient of axion couples to nucleon is similar to the way that
magnetic field couples to nucleon spin. Using magnetometers, Bloch et al
placed constraints on axion-neutron axial-vector coupling for a wide range of
axion mass, $10^{-24}\,{\rm eV}\lesssim m\lesssim 10^{-13}\,{\rm eV}$ [36,
37]. The constraint on $f$ (gray shaded) is similar to the one from Iodine
molecular spectroscopy, while future projection (gray dashed) could dominate
the supernova bound. In addition, axion-nucleon coupling can be searched with
nuclear magnetic resonance (NMR) [57, 58]. Current constraint using NMR
technique is $f\gtrsim\,{\rm GeV}$ at $m\simeq 10^{-7}\,{\rm eV}$ [39], while
future projection with a much larger sample size (dark blue dashed) might
reach to the QCD axion parameter space.

It is interesting to compare the reach of spectroscopic experiments with
astrophysical and cosmological constraints. Axion-nucleon couplings provide an
additional energy-loss channel for stellar objects. Considering the
observation of neutrino flux from supernova SN1987A, the axion-nucleon
coupling is constrained as $f\gtrsim 4\times 10^{8}\,{\rm GeV}$ [35] (see also
alternative claim by Bar et al [59]). Neutron star provides additional
constraints on axion-nucleon coupling [60, 61, 62, 63, 64, 65, 66]. A recent
study on the cooling of old neutron stars provides $f\gtrsim 3\times
10^{8}\,{\rm GeV}$ [66], which is comparable to the supernova bound.

Another interesting bound, which is not shown in Figure 1, is the constraint
from big bang nucleosynthesis (BBN) [67]. Blum et al took the cosmic dark
matter density in the present universe, extrapolated the field value to the
value at the time of BBN assuming the standard cosmological evolution of
axion, $\theta(t_{\rm BBN})\simeq\theta(t_{0})\min[(a_{0}/a_{\rm
BBN})^{3/2},(a_{0}/a_{m})^{3/2}]$ with $a_{m}$ being scale factor at $3H=m$,
and computed the modification of the helium 4 mass fraction due to the
neutron/proton mass difference arising from axion field [67]. This bound
constrains a large part of parameter space for the mass range shown in the
figure. The BBN bound might be avoided if, for some reason, the extrapolation
fails and $\theta(t_{\rm BBN})\lesssim 1$. The deviation of proton-neutron
mass difference from its standard value becomes quickly negligible for
moderately small $\theta$ [68].

In summary, we have shown that the axion-gluon coupling induces an oscillation
in nuclear quantities in the presence of axion dark matter and that it can be
probed by atomic and molecular spectroscopy as well as interferometry
experiments. Similarly to dilaton-like DM [13] or relaxion DM [14, 32], atomic
energy levels oscillate due to the axion DM background, leaving harmonic
signals in the transition frequencies of stable frequency standards. By using
existing experimental searches for scalar DM-SM interactions, we are able to
place constraints on the axion-gluon interaction. Current constraints are
generally weaker than existing constraints such as oscillating neutron EDM,
but some of them already compete with astrophysical bounds e.g. SN1987A
constraint. Future experiments based on nuclear clock transition or molecular
vibrational excitation are expected to probe directly and model-independently
axion model in a region of parameter space that is currently unexplored.

Note added While this work has been finalized, a related work [69] appeared on
arXiv, where the authors investigated the implications of the cosmological
variation of $\bar{\theta}$.

###### Acknowledgements.

We would like to thank Abhishek Banerjee, Fady Bishara, Joshua Eby, Zhen Liu,
Oleksii Matsedonskyi, Ethan Neil, Pablo Quilez, and Seokhoon Yun for useful
discussions. We also thank Anson Hook for useful comments on the manuscript.
The work of HK was supported by the Deutsche Forschungsgemeinschaft under
Germany’s Excellence Strategy - EXC 2121 Quantum Universe - 390833306 and by
the Munich Institute for Astro- and Particle Physics (MIAPP) which is funded
by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy –
EXC-2094 – 390783311. The work of GP is supported by grants from BSF-NSF,
Friedrich Wilhelm Bessel research award, GIF, ISF, Minerva, SABRA - Yeda-Sela
- WRC Program, the Estate of Emile Mimran, and the Maurice and Vivienne Wohl
Endowment.

##  Appendix A $\theta$-dependence of nuclear quantities

We detail the $\theta$-dependence of nuclear quantities.

The nucleon mass at the one-loop or chiral order ${\cal O}(p^{3})$ is [70, 71,
72]

$\displaystyle
m_{N}(\theta)=m_{0}-4c_{1}m_{\pi}^{2}(\theta)-\frac{3g_{A}^{2}m_{\pi}^{3}(\theta)}{32\pi
f_{\pi}^{2}}$ (8)

where $m_{0}$ is the nucleon mass in the chiral limit, $g_{A}=1.27$ is the
axial-vector coupling, and $c_{1}=-1.1\,{\rm GeV}^{-1}$ is a low energy
constant [73]. Therefore, we find

$\displaystyle\frac{\partial\ln m_{N}}{\partial\ln m_{\pi}^{2}}\simeq 0.06,$
(9)

where we have used $f_{\pi}=93\,{\rm MeV}$, and $m_{N}=939\,{\rm MeV}$.
Alternatively, we can write $\partial\ln m_{N}/\partial\ln
m_{\pi}^{2}=(\sigma_{\pi N}/m_{N})$. Current lattice computations predict
$30\,{\rm MeV}\lesssim\sigma_{\pi N}\lesssim 70\,{\rm MeV}$ (see review [74]),
which translates into $0.03\lesssim\partial\ln m_{N}/\partial\ln
m_{\pi}\lesssim 0.08$. We use Eq. (9) for the analysis in the main text.

The nucleon $g$-factor can also be computed from chiral Lagrangian. At the
chiral order ${\cal O}(p^{3})$, one finds [72]

$\displaystyle g_{p}(\theta)$ $\displaystyle=$ $\displaystyle
g_{p}^{(0)}-\frac{g_{A}^{2}m_{N}m_{\pi}(\theta)}{4\pi f_{\pi}^{2}},$ (10)
$\displaystyle g_{n}(\theta)$ $\displaystyle=$ $\displaystyle
g_{n}^{(0)}+\frac{g_{A}^{2}m_{N}m_{\pi}(\theta)}{4\pi f_{\pi}^{2}}.$ (11)

Therefore, we find

$\displaystyle\frac{\partial\ln g_{p}}{\partial\ln m_{\pi}^{2}}$
$\displaystyle=$
$\displaystyle-\frac{1}{g_{p}}\frac{g_{A}^{2}m_{N}m_{\pi}}{8\pi
f_{\pi}^{2}}\simeq-0.17$ (12) $\displaystyle\frac{\partial\ln
g_{n}}{\partial\ln m_{\pi}^{2}}$ $\displaystyle=$
$\displaystyle+\frac{1}{g_{n}}\frac{g_{A}^{2}m_{N}m_{\pi}}{8\pi
f_{\pi}^{2}}\simeq-0.25$ (13)

where we have used $g_{n}=-3.826$ and $g_{p}=5.586$. The actual value might be
smaller than the above values. One might extract isovector magnetic moment
from form factors computed by lattice simulation [75], and the resulting
dependence of $g$-factor on pion mass from such result is a factor two or
three smaller than Eqs. (12)–(13). For the purpose of order of magnitude
estimation, we use Eqs. (12)–(13).

The nuclear $g$-factor can be written as a function of nucleon $g$-factor as
well as spin expectation value of valence and core nucleons. Following Ref.
[41], one finds

$\frac{\delta g}{g}=\left[K_{n}\frac{\partial\ln g_{n}}{\partial\ln
m_{\pi}^{2}}+K_{p}\frac{\partial\ln g_{p}}{\partial\ln
m_{\pi}^{2}}-0.17K_{b}\right]\frac{\delta m_{\pi}^{2}}{m_{\pi}^{2}}$

With values of $K_{n,p,b}$ given in [41], we find $\delta g/g=-0.02(\delta
m_{\pi}^{2}/m_{\pi}^{2})$ for 87Rb and $\delta g/g=0.01(\delta
m_{\pi}^{2}/m_{\pi}^{2})$ for 133Cs.

The variation of nuclear clock transition in 299Th is estimated in Refs. [45,
46].

$\displaystyle\frac{\delta f_{\rm Th}}{f_{\rm Th}}$ (14) $\displaystyle\simeq$
$\displaystyle 1.3\times 10^{5}\left(-12\frac{\delta
m_{N}}{m_{N}}+10\frac{\delta m_{\Delta}}{m_{\Delta}}+6\frac{\delta
m_{\pi}}{m_{\pi}}-43\frac{\delta m_{V}}{m_{V}}\right)$ $\displaystyle\simeq$
$\displaystyle 2\times 10^{5}\frac{\delta m_{\pi}^{2}}{m_{\pi}^{2}},$

where $m_{\Delta}$ and $m_{V}$ are the masses of $\Delta$-baryon and vector
meson. Among contributions from different mesons and hadrons, the pion
contribution dominates all. It is straightforward to check that

$\displaystyle\frac{\delta m_{N}}{m_{N}}$ $\displaystyle\simeq$ $\displaystyle
0.13\frac{\delta m_{\pi}}{m_{\pi}}$ (15) $\displaystyle\frac{\delta
m_{\Delta}}{m_{\Delta}}$ $\displaystyle\simeq$ $\displaystyle 0.03\frac{\delta
m_{\pi}}{m_{\pi}}$ (16) $\displaystyle\frac{\delta m_{V}}{m_{V}}$
$\displaystyle\simeq$ $\displaystyle 0.04\frac{\delta m_{\pi}}{m_{\pi}}$ (17)

where the variance of $\Delta$ baryon can be obtained directly from Eq. (62)
in [76] or from $\sigma_{\pi\Delta}=m_{\pi}^{2}\partial m_{\Delta}/\partial
m_{\pi}^{2}=20.6\,{\rm MeV}$. The variation of vector meson mass is obtained
in [45, 77].

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