Obviously, the 4-dimensional effective gravitational and Yang-Mills couplings depend on the considered superstring theory, on the compactification scheme but in any case they depend on the dilaton.

As an example, Maeda (1988) considered the ($N=1, D=10$)-supergravity model derived from the heterotic superstring theory in the low energy limit and assumed that the 10-dimensional spacetime is compactified on a 6-torus of radius $R(x^\mu)$ so that the effective 4-dimensional theory described by (75) is of the Brans-Dicke type with $\omega = -1$. Assuming that $\phi$ has a mass $\mu$, and couples to the matter fluid in the universe as $S_{\text{matter}} = \int d^{10}\mathbf{x}\sqrt{-g_{10}}\exp(-2\Phi)\mathcal{L}_{\text{matter}}(g_{10})$, the reduced 4-dimensional matter action is

$$S_{\text{matter}} = \int d^4\mathbf{x}\sqrt{-g}\phi\mathcal{L}_{\text{matter}}(g). \qquad (82)$$

The cosmological evolution of $\phi$ and $R$ can then be computed to deduce that $\alpha_{\text{EM}}/\alpha_{\text{EM}} \simeq 10^{10} (\mu/1 \text{eV})^{-2} \text{yr}^{-1}$. Vayonakis (1988) considered the same model but assumed that supersymmetry is broken by non-perturbative effects such as gaugino condensation. Hence, and contrary to Maeda (1988), $\phi$ is stabilized and the variation of the constants arises mainly from the variation of $R$ in a runaway potential.

Kiritsis (1999) considered a probe D3-brane in the context of AdS/CFT correspondence at finite temperature and provides the predictions for the running electric and magnetic effective couplings, beyond perturbation theory. It allows to construct a varying speed of light model.

To conclude, superstring theories offer a natural theoretical framework to discuss the value of the fundamental constants since they become expectation values of some fields. This is a first step towards their understanding but yet, no complete and satisfactory mechanism for the stabilization of the extra-dimensions and dilaton is known. It has paved the way for various models that we detail in Sect. 7.

## 4 Relations between constants

As soon as one consider unification schemes, various constants can exhibit correlated variations, which indeed are expected to be model-dependent. The second theoretical aspect one shall consider is the possible correlation between the variations of several fundamental constants and the dependencies of primary parameters on fundamental constants. This requires to have a deeper look on the matter sector, i.e., on the standard model of particle physics and its extensions.

First, in quantum field theory, one needs to take into account the running of coupling constants with energy and the possibilities of grand unification to bind them so that their variations will derive from the one of a single high-energy coupling. It will give a link between the QCD scale, the coupling constants and the masses of the fundamental particles (i.e., the Yukawa couplings and the Higgs vev), as described in Sect. 4.1. Second, in order to relate fundamental constant to the nuclear primary parameters, one compute the binding energies and the masses of the proton, neutron and different nuclei in terms of the gauge couplings and the quark
masses. This step involves QCD and nuclear physics. Similarly, one can express the gyromagnetic factors in terms of the quark masses; see Sect. 4.3. This step, described in Sect. 4.2, is particularly important to interpret the constraints from the atomic clocks and the QSO spectra. In particular, it allows one to set stronger constraints on the varying parameters at the expense of a model-dependence. To finish, we present in Sect. 4.5 the phenomenological (R, S)-parameterisation.

## 4.1 Implication of gauge coupling unification

The first theoretical implication of high-energy physics arises from the unification of the non-gravitational interactions. In these unification schemes, the three standard model coupling constants derive from one unified coupling constant so that their variation shall be correlated.

### Renormalisation and running of the couplings constants

In quantum field theory, the calculation of scattering processes include higher order corrections of the coupling constants related to loop corrections that introduce integrals over internal 4-momenta. Depending on the theory, these integrals may be either finite or diverging as the logarithm or power law of a UV cut-off. In a class of theories, called renormalizable, among which the standard model of particle physics, the physical quantities calculated at any order do not depend on the choice of the cut-off scale. But the result may depend on $\ln E/m$ where E is the typical energy scale of the process. It follows that the values of the coupling constants of the standard model depend on the energy at which they are measured (or of the process in which they are involved). This running arises from the screening due to the existence of virtual particles, which are polarized by the presence of a charge. The renormalization group allows one to compute the dependence of a coupling constants on the energy E as

$$ \frac{\mathrm{d}g_i(E)}{\mathrm{d} \ln E} = \beta_i(E), $$

where the beta functions, $\beta_i$, depend on the gauge group and on the matter content of the theory and may be expended in powers of $g_i$. For the SU(2) and U(1) gauge couplings of the standard model, they are given by

$$ \beta_2(g_2) = -\frac{g_2^3}{4\pi^2}\left(\frac{11}{6} - \frac{n_g}{3}\right), \quad \beta_1(g_1) = +\frac{g_1^3}{4\pi^2}\frac{5n_g}{9} $$

where $n_g$ is the number of generations for the fermions. We remind that the fine-structure constant is defined in the limit of zero momentum transfer so that cosmological variation of $\alpha_{EM}$ are independent of the issue of the renormalization group dependence. For the SU(3) sector, with fundamental Dirac fermion representations,
$$ \beta_3(g_3) = - \frac{g_3^3}{4\pi^2} \left( \frac{11}{4} - \frac{n_f}{6} \right), $$

$n_f$ being the number of quark flavors with mass smaller than $E$. The negative sign implies that (1) at large momentum transfer the coupling decreases and loop corrections become less and less significant: QCD is said to be asymptotically free; (2) integrating the renormalization group equation for $\alpha_3$ gives

$$ \alpha_3(E) = \frac{6\pi}{(33 - n_f) \ln(E/\Lambda_c)} $$

so that it diverges as the energy scale approaches $\Lambda_c$ from above, that we decided to call $\Lambda_{QCD}$. This scale characterizes all QCD properties and in particular the masses of the hadrons are expected to be proportional to $\Lambda_{QCD}$ up to corrections of order $m_q/\Lambda_{QCD}$.

## Unification

It was noticed quite early that these relations imply that the weaker gauge coupling becomes stronger at high energy, while the strong coupling becomes weaker so that one can think the three non-gravitational interactions may have a single common coupling strength above a given energy. This is the driving idea of Grand Unified Theories (GUT) in which one introduces a mechanism of symmetry-breaking from a higher symmetry group, such, e.g., as SO(10) or SU(5), at high energies.

It has two important consequences for our present considerations. First there may exist algebraic relations between the Yukawa couplings of the standard model. Second, the structure constants of the standard model unify at an energy scale $m_u$

$$ \alpha_1(m_u) = \alpha_2(m_u) = \alpha_3(m_u) \equiv \alpha_U(m_u). \qquad (83) $$

We note that the electroweak mixing angle can also be a time dependent parameter, but only for $E \neq m_u$ since at $E = m_u$, it is fixed by the symmetry to have the value $\sin^2 \theta = 3/8$, from which we deduce

$$ \alpha_{\text{EM}}^{-1}(M_Z) = \frac{5}{3} \alpha_1^{-1}(M_Z) + \alpha_2^{-1}(M_Z). $$

It follows from the renormalization group relations that

$$ \alpha_i^{-1}(E) = \alpha_i^{-1}(m_u) - \frac{b_i}{2\pi} \ln \frac{E}{m_u}, \qquad (84) $$

where the beta-function coefficients are given by $b_i = (41/10, -19/6, 7)$ for the standard model (or below the SUSY scale $\Lambda_{\text{SUSY}}$) and by $b_i = (33/5, 1, -3)$ for $N=1$ supersymmetric theory. Given a field decoupling at $m_{\text{th}}$, one has
$$ \alpha_i^{-1}(E_-) = \alpha_i^{-1}(E_+) - \frac{b_i^{(-)}}{2\pi} \ln \frac{E_-}{E_+} - \frac{b_i^{(th)}}{2\pi} \ln \frac{m_{th}}{E_+} $$

where $b_i^{(th)} = b^{(+)} - b^{(-)}$ with $b^{(+/-)}$ the beta-function coefficients respectively above and below the mass threshold, with tree-level matching at $m_{th}$. In the case of multiple thresholds, one must sum the different contributions. The existence of these thresholds implies that the running of $\alpha_3$ is complicated since it depends on the masses of heavy quarks and colored superpartner in the case of supersymmetry. For non-supersymmetric theories, the low-energy expression of the QCD scale is

$$ \Lambda_{QCD} = E \left( \frac{m_c m_b m_t}{E} \right)^{2/27} \exp \left( - \frac{2\pi}{9\alpha_3(E)} \right) \quad (85) $$

for $E > m_t$. This implies that the variation of Yukawa couplings, gauge couplings, Higgs vev and $\Lambda_{QCD}/M_P$ are correlated.

A second set of relations arises in models in which the weak scale is determined by dimensional transmutation (Ellis et al. 1982a, b). In such cases, the Higgs vev is related to the Yukawa constant of the top quark by (Campbell and Olive 1995)

$$ v = M_P \exp \left( - \frac{8\pi^2 c}{h_t^2} \right), \quad (86) $$

where $c$ is a constant of order unity. This would imply that

$$ \delta \ln v = S \delta \ln h \quad (87) $$

with $S \sim 160$ (Coc et al. 2007).

### Consequences for the coupled variation of fundamental constants

The first consequences of unification were investigated in Ellis et al. (1982b), Campbell and Olive (1995), Calmet and Fritzsch (2002a, 2002b), Damour and Polyakov (1994a, 1994b), Langacker et al. (2002) where the variation of the 3 coupling constants was reduced to the one of $\alpha_U$ and $m_u/M_P$. It was concluded that, setting

$$ R \equiv \delta \ln \Lambda_{QCD} / \delta \ln \alpha_{EM}, \quad (88) $$

$R \sim 34$ with a stated accuracy of about 20% (Langacker et al. 2002; Langacker 2004) (assuming only $\alpha_U$ can vary), $R \sim 40.82$ in the string dilaton model assuming Grand Unification (Damour and Polyakov 1994a, b) (see Sect. 7.1), $R = 38 \pm 6$ (Calmet and Fritzsch 2002b) and then $R = 46$ (Calmet and Fritzsch 2002a, 2006), the difference arising from the quark masses and their associated thresholds. However, these results implicitly assume that the electroweak symmetry breaking and supersymmetry breaking mechanisms, as well as the fermion mass generation, are not affected by the variation of the unified coupling. It was also mentioned in Calmet and Fritzsch (2002a) that $R$ can reach $-235$ in unification based on SU(5) and SO(10). The large value of $R$ arises from the exponential dependence of $\Lambda_{QCD}$ on $\alpha_3$. In the limit in which the quark masses are set to zero, the proton mass, as well as all other hadronic masses are proportional to $\Lambda_{QCD}$, i.e.,
$m_p \propto \Lambda_{QCD}[1 + O(m_q/\Lambda_{QCD})]$. Langacker et al. (2002) further related the Higgs vev to $\alpha_{EM}$ by $d \ln \nu/d \ln \alpha_{EM} \equiv \kappa$ and estimated that $\kappa \sim 70$ so that, assuming that the variation of the Yukawa couplings is negligible, it could be concluded that

$$ \delta \ln \frac{m}{\Lambda_{QCD}} \sim 35 \delta \ln \alpha_{EM}, $$

for the quark and electron masses. This would also imply that the variations of $\mu$ and $\alpha_{EM}$ are correlated, still in a very model-dependent way, typically one can conclude (Coc et al. 2007) that

$$ \frac{\delta \mu}{\mu} = -0.8R \frac{\delta \alpha_{EM}}{\alpha_{EM}} + 0.6(S + 1) \frac{\delta h}{h}, $$

with $S \sim 160$. The running of $\alpha_U$ can be extrapolated to the Planck mass, $M_P$. Assuming $\alpha_U(M_P)$ fixed and letting $m_u/M_P$ vary, it was concluded (Dine et al. 2003) that $R = 2\pi(b_U + 3)/[9\alpha_{EM}(8b_U/3 - 12)]$ where $b_U$ is the beta-function coefficient describing the running of $\alpha_U$. This shows that a variation of $\alpha_{EM}$ and $\mu$ can open a windows on GUT theories. A similar analysis (Dent 2004) assuming that electroweak symmetry breaking was triggered by non-perturbative effects in such a way that $\nu$ and $\alpha_U$ are related, concludes that $\delta\mu/\mu = (13 \pm 7)\delta\alpha_{EM}/\alpha_{EM}$ in a theory with soft SUSY breaking and $\delta\mu/\mu = (-4 \pm 5)\delta\alpha_{EM}/\alpha_{EM}$ otherwise.

From a phenomenological point of view, Dent et al. (2008) assumed a proportionality with fixed “unification coefficients” and that the variations of the constants at a given redshift $z$ depend on a unique evolution factor $\ell(z)$ and can be derived from those of the unification mass scale (in Planck units), $m_u$, the unified gauge coupling $\alpha_U$, the Higgs vev, $\nu$ and in the case of supersymmetric theories the soft supersymmetry breaking mass, $\tilde{m}$. Introducing the coefficients $d_i$ by

$$ \Delta \ln \frac{m_u}{M_P} = d_M \ell, \quad \Delta \ln \alpha_U = d_U \ell, \quad \Delta \ln \frac{\nu}{m_u} = d_H \ell, \quad \Delta \ln \frac{\tilde{m}}{M_P} = d_S \ell, $$

$(d_S = 0$ for non-supersymmetric theories) and assuming that the masses of the standard model fermions all vary with $\nu$ so that the Yukawa couplings are assumed constant, it was shown that the variations of all constants can be related to $(d_M, d_U, d_H, d_S)$ and $\ell(z)$, using the renormalization group equations (neglecting the effects induced by the variation of $\alpha_U$ on the RG running of fermion masses). This decomposition is a good approximation provided that the time variation is slow, which is actually backed up by the existing constraints, and homogeneous in space (so that it may not be applied as such in the case a chameleon mechanism is at work (Brax and Martin 2007b).

This allowed to be defined six classes of scenarios: (1) varying gravitational constant ($d_H = d_S = d_U = 0$) in which only $m_u/M_P$ or equivalently $G\Lambda_{QCD}^2$ is varying; (2) varying unified coupling ($d_U = 1, d_H = d_S = d_M = 0$); (3) varying Fermi scale defined by ($d_H = 1, d_U = d_S = d_M = 0$) in which one has $d \ln \mu/d \ln \alpha_{EM} = -325$; (4) varying Fermi scale and SUSY-breaking scale ($d_S = d_H = 1, d_U = d_M = 0$) and for which $d \ln \mu/d \ln \alpha_{EM} = -21.5$; (5) varying unified coupling and Fermi scale ($d_U = 1, d_H = \tilde{\gamma}d_U, d_S = d_M = 0$) and for which $d \ln \mu/d \ln \alpha_{EM} = (23.2 - 0.65\tilde{\gamma})/(0.865 + 0.02\tilde{\gamma})$; (6) varying unified coupling
and Fermi scale with SUSY ($d_U = 1, d_S \simeq d_H = \tilde{\gamma}d_U, d_M = 0$) and for which $d \ln \mu/d \ln \alpha_{EM} = (14 - 0.28\tilde{\gamma})/(0.52 + 0.013\tilde{\gamma})$.

Using the dependence of $m_p$ on the Higgs vev (Gasser and Leutwyler 1982), Mohamadnejad (2019), Chakrabarti (2021) deduced that $\Delta\mu/\mu = -0.91\Delta\nu/\nu$ in order to constrain the cosmological variation of $\nu$ assuming its dynamics is incorporated from a Brans-Dicke action.

Each scenario can be compared to the existing constraints to get sharper bounds on them (Dent et al. 2007, 2008, 2009; Müller et al. 2004) and emphasize that the correlated variation between different constants (here $\mu$ and $\alpha_{EM}$) depends strongly on the theoretical hypothesis that are made.

### Effect of the QCD vacuum angle $\theta$

Motivated by the naturalness problem of the QCD vacuum angle $\theta$, whose natural order of magnitude would be of order unity while it is experimentally observed to be smaller than $10^{-10}$ (Abel et al. 2020). The fact that, contrary to the cosmological constant, there seem to be no anthropic limitations on $\theta$ motivated Lee et al. (2020) to investigate the impact of the QCD vacuum on light nuclei starting. The question of the $\theta$-dependence of nuclear physics was addressed by Ubaldi (2010) considering the effect of the $\theta$ of the QCD Lagrangian

$$ \mathcal{L}_{\theta} = -\frac{g^2\theta}{32\pi^2} F\tilde{F} \qquad (89) $$

which leads to CP-violation in the strong interaction. It concluded that the binding energies of deuteron and diproton would change by 10% for $\theta \sim 130^\circ$, which could affect BBN and that for $\theta \sim 2^\circ$ the effects on the 3$\alpha$-process would imply that the abundances of carbon and oxygen would be ten times greater. This led Lee et al. (2020) to investigate the impact of the QCD vacuum on BBN and the 3$\alpha$ process. They started from the nucleon mass dependence (Brower et al. 2003)

$$ m_n = m_0 - 4c_1 M_{\pi}^2(\theta) - \frac{3g_A^2}{32\pi F_{\pi}^2} M_{\pi}^3(\theta) \qquad (90) $$

where $m_0 \simeq 865$ MeV is the nucleon mass in the chiral limit, $g_A = 1.27$ the axial-vector coupling constant, $F_\pi$ the pion decay constant and $c_1$ a low-energy constant from the second order chiral pion-nucleon and where the pion mass behaves as

$$ M_{\pi}^{2}(\theta) = M_{\pi}^{2} \cos \frac{\theta}{2} \sqrt{1 + \varepsilon^{2} \tan^{2} \frac{\theta}{2}} $$

with $M_\pi = 139.57$ MeV the charged pion mass and $\varepsilon = (m_u - m_d)/(m_u + m_d)$ and the $\theta$-dependence of the pion-nucleon coupling $g_{\pi NN}(\theta)$ to compute the neutron-proton mass difference

$$ (m_n - m_p)^{\text{QCD}}(\theta) = 4c_5 B_0 \frac{M_{\pi}^2}{M_{\pi}^2(\theta)} (m_u - m_d) \qquad (91) $$

and show that its magnitude increases with $\theta$ reaching 1% at $\theta \sim 1$. Similarly they
showed that the neutron lifetime drops off very quickly when $\theta$ deviates from 0 so that effects on BBN are expected to be important. This allowed them to investigate the effects on primordial and stellar nucleosynthesis to conclude that the dineutron and diproton are bound for $\theta \ge 0.2$ and $\theta \ge 0.7$ and that $\theta$ must be smaller than 0.1 to recover the real nuclear reaction rates, that can then be used safely since experimentally, $\theta < 10^{-10}$.

This line of analysis was followed by Kim and Perez (2024) who considered the coupling of the gluon to an axion $a$ so that Eq. (89) is modified to $\mathcal{L} \propto aF\tilde{F}$. For an ultralight dark matter field, its oscillations will induce fluctuations of all nuclear quantities. Using the dependence (90) for the nucleon mass, and

$$g_p = g_p^{(0)} - \frac{g_A^2}{4\pi F_{\pi}^2} m_n M_{\pi}(\theta) \qquad g_n = g_n^{(0)} + \frac{g_A^2}{4\pi F_{\pi}^2} m_n M_{\pi}(\theta)$$

for the proton and nucleon gyromagnetic factors. Following Flambaum and Tedesco (2006) they computed the $\theta$-dependence for thorium and the standard atomic clocks; see Kim and Perez (2024) for a similar analysis. Davoudiasl et al. (2022) built on this analysis to relate a variation of $\theta$ to a variation of $\bar{\mu}$ as $\Delta(\theta^2) \sim -1.4\Delta\bar{\mu}/\bar{\mu}$ to conclude from the constraints on the variation of $\bar{\mu}$ thanks to the Yb-clocks experiment (Lange et al. 2021) that

$$\frac{d\theta^2}{dt} < 6 \times 10^{-15} \text{ yr}^{-1}. \qquad (92)$$

Flambaum and Samsonov (2023) considered an ultralight dark matter axion $\phi = \theta/f_a$. Using the results from Flambaum and Tedesco (2006), Kim and Perez (2024) they concluded from the analysis of Rb-Cs experiment by Hees et al. (2016) that the axion decay constant shall satisfy $f_a > 1.8 \times 10^9 \text{ GeV}(10^{-15} \text{ eV}/m_\phi)$. Then, from the Rb-Cs clock experiment, Dzuba et al. (2024) deduced that

$$\frac{d\theta^2}{dt} = (8.2 \pm 16) \times 10^{-13} \text{ yr}^{-1}. \qquad (93)$$

Similar analysis for Al/Hg gave $(1.8 \pm 2.8) \times 10^{-13} \text{ yr}^{-1}$ while those for Yb-clocks (Banerjee et al. 2023a) and Rb-quartz and Dy (Zhang et al. 2023) allowed to constrain linear and quadratic ultralight scalar and axion dark matter models.

## 4.2 Masses and binding energies in terms of the fundamental constants

For “composite” systems such as proton, neutron, nuclei or even planets and stars, we need to compute their mass, which requires to determine their binding energies. As already seen, the electromagnetic binding energy induces a direct dependence on $\alpha_{EM}$ and can be evaluated using, e.g., the Bethe–Weizsäcker formula (28). The dependence of the masses on the quark masses, via nuclear interactions, and the determination of the nuclear binding energy are especially difficult to estimate.
### Nucleon mass

In the chiral limit of QCD, in which all quark masses are negligible compared to $\Lambda_{QCD}$, all dimensionful quantities scale as some power of $\Lambda_{QCD}$. For instance, concerning the nucleon mass, $m_N = c\Lambda_{QCD}$ with $c \sim 3.9$ being computed from lattice QCD. This predicts a mass of order 860 MeV, smaller than the observed value of 940 MeV. The nucleon mass can be computed in chiral perturbation theory and expressed in terms of the pion mass as (Leinweber et al. 1999) $m_N = a_0 + a_2m_\pi^2 + a_4m_\pi^4 + a_6m_\pi^6 + \sigma_{N\pi} + \sigma_{A\pi} + \sigma_{tad}$ (where all coefficients of this expansion are defined in Leinweber et al. 1999), which can be used to show (Flambaum et al. 2004) that the nucleon mass scales as

$$m_N \propto \Lambda_{QCD} X_q^{0.037} X_s^{0.011} \qquad (94)$$

with the definition (3) for the quantities $X$. It follows that

$$\bar{\mu} \propto X_q^{-0.037} X_s^{-0.011} X_e \qquad (95)$$

so that when one assumes $X_s \propto X_q$ it reduces to

$$\bar{\mu} \propto X_q^{-0.048} X_e. \qquad (96)$$

Note, however, that such a notation is dangerous since it would imply that $m_N$ vanishes in the chiral limit but it is a compact way to give $\delta m_N/\delta X_q$ etc. It was further extended (Flambaum and Shuryak 2003) by using a sigma model to infer that $m_N \propto \Lambda_{QCD} X_q^{0.045} X_s^{0.19}$. These two examples explicitly show the strong dependence in nuclear modeling.

### Deuterium binding energy, $B_D$

Before discussing the general case, let us consider the expression of the deuterium binding energy $B_D$ has been discussed in different ways (see Sect. 5.14.4). Many models have been created.

A first route relies on the use of the dependence of $B_D$ on the pion mass (Epelbaum et al. 2003; Beane and Savage 2003; Pudliner et al. 1997; Yoo and Scherrer 2003), which can then be related to $m_u$, $m_d$ and $\Lambda_{QCD}$.

A second avenue is to use a sigma model in the framework of the Walecka model (Serot and Walecka 1997) in which the potential for the nuclear forces keeps only

<table><caption>**Table 5** Sensitivity coefficients of a variation of the binding energies of light elements on a variation of ($\alpha_{EM}$, $m_q$); see Eq. (100)</caption><thead><tr><th>X</th><th>c<sub>x</sub></th><th>c<sub>q</sub></th></tr></thead><tbody><tr><td>tritium</td><td>-0.047</td><td>-2.1</td></tr><tr><td>helium-3</td><td>-0.093</td><td>-2.3</td></tr><tr><td>helium-4</td><td>-0.030</td><td>-0.94</td></tr><tr><td>lithium-7</td><td>-0.046</td><td>-1.4</td></tr><tr><td>berrilium-7</td><td>-0.089</td><td>-1.4</td></tr></tbody></table>
the $\sigma$, $\rho$ and $\omega$ meson exchanges (Flambaum and Shuryak 2003). We also emphasize that the deuterium is only produced during BBN, as it is too weakly bound to survive in the regions of stars where nuclear processes take place.

The fact that we do observe deuterium today sets a non-trivial constraint on the constants by imposing that the deuterium remains stable from BBN time to today. Since it is weakly bound, it is also more sensitive to a variation of the nuclear force compared to the electromagnetic force. This was used in Dent and Fairbairn (2003) to constrain the variation of the nuclear strength in a sigma-model.

### Atomic nuclei masses

To go further and determine the sensitivity of the mass of a nucleus to the various constant,

$$m(A, Z) = Zm_p + (A - Z)m_n + Zm_e + E_S + E_{EM}$$

one should determine the strong binding energy [see related discussion below Eq. (30)] in function of the atomic number Z and the mass number A.

While the particular case of deuterium has been discussed in the previous paragraph, the situation is more complicated for larger nuclei since there is no simple modeling. For large mass number A, the strong binding energy can be approximated by the liquid drop model

$$\frac{E_S}{A} = a_V - \frac{a_S}{A^{1/3}} - a_A \frac{(A - 2Z)^2}{A^2} + a_P \frac{(-1)^A + (-1)^Z}{A^{3/2}} \quad (97)$$

with $(a_V, a_S, a_A, a_P) = (15.7, 17.8, 23.7, 11.2)$ MeV (Rohlf 1994). It has also been suggested (Damour and Lilley 2008) that the nuclear binding energy can be expressed as

$$E_S \approx Aa_3 + A^{2/3}b_3 \quad \text{with} \quad a_3 = a_3^{\text{chirallimit}} + m_{\pi}^2 \frac{\partial a_3}{\partial m_{\pi}^2}. \quad (98)$$

In the chiral limit, $a_3$ has a non-vanishing limit to which we need to add a contribution scaling like $m_{\pi}^2 \propto \Lambda_{QCD}m_q$. Damour and Lilley (2008) also pointed out that the delicate balance between attractive and repulsive nuclear interactions (Serot and Walecka 1997) implies that the binding energy of nuclei is expected to depend strongly on the quark masses (Donoghue 2006). A fitting formula derived from effective field theory and based of the semi-empirical formula derived in Furnstahl and Serot (2000) was also proposed (Damour and Donoghue 2008) as

$$\frac{E_S}{A} = -\left(120 - \frac{97}{A^{1/3}}\right)\eta_S + \left(67 - \frac{57}{A^{1/3}}\right)\eta_V + \dots \quad (99)$$

where $\eta_S$ and $\eta_V$ are the strength of respectively the scalar (attractive) and vector (repulsive) nuclear contact interactions normalized to their actual value. These two parameters need to be related to the QCD parameters (Donoghue 2006).

We refer to Flambaum and Wiringa (2007) for the study of the dependence of the binding of light ($A \le 8$) nuclei on possible variations of hadronic masses, including
meson, nucleon, and nucleon-resonance masses. The following approximate relations have been widely used in BBN analysis (Flambaum and Wiringa 2007; Dent et al. 2007)

$$ \Delta B_X / B_X = c_\alpha \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}} + c_q \Delta m_q / m_q \quad (100) $$

with the sensitivity coefficients summarized in Table 5.

These expressions allow one to compute the sensitivity coefficients that enter in the decomposition of the mass; see Eq. (342). They stress one of the most difficult issue related to the intricate structure of QCD and its role in low energy nuclear physics, which is central to determine the masses of nuclei and the binding energies, quantities that are particularly important for BBN, the universality of free fall and stellar physics.

## 4.3 Gyromagnetic factors in terms of the fundamental constants

Since they enter in the constraints arising from the comparison of atomic clocks and QSO, it important to relate the gyromagnetic factors to fundamental constants.

### Proton and neutron gyromagnetic factors

The proton and neutron gyromagnetic factors are respectively given by $g_p = 5.586$ and $g_n = -3.826$ and are expected to depend on $X_q = m_q / \Lambda_{QCD}$ (Flambaum 2003). In the chiral limit in which $m_u = m_d = 0$, the nucleon magnetic moments remain finite so that one could have thought that the finite quark mass effects should be small. However, it is enhanced by $\pi$-meson loop corrections, which are proportional to $m_\pi \propto \sqrt{m_q \Lambda_{QCD}}$. Following Leinweber et al. (1999), this dependence can be described by the approximate formula

$$ g(m_\pi) = \frac{g(0)}{1 + am_\pi + bm_\pi^2} $$

The coefficients are given by $a = (1.37, 1.85) / \text{GeV}$ and $b = (0.452, 0.271) / \text{GeV}^2$ respectively for the proton and neutron. This lead Flambaum (2003) to $g_p \propto m_\pi^{-0.174} \propto X_q^{-0.087}$ and $g_n \propto m_\pi^{-0.213} \propto X_q^{-0.107}$. This was further extended in Flambaum et al. (2004) to take into account the dependence with the strange quark mass $m_s$ to obtain

$$ g_p \propto X_q^{-0.087} X_s^{-0.013}, \quad g_n \propto X_q^{-0.118} X_s^{0.0013} \quad (101) $$

with the definitions (3). All these expressions assume $\Lambda_{QCD}$ constant in their derivations (Flambaum 2004; Flambaum and Tedesco 2006; Flambaum and Dzuba 2009; Dinh et al. 2009; Luo et al. 2011; Jackson Kimball 2015). In the approximation $X_s \propto X_q$ which is well-motivated by the Higgs mechanism of mass generation, the dependence in the quark mass reduces to

$$ g_p \propto X_q^{-0.10}, \quad g_n \propto X_q^{-0.1167}. \quad (102) $$
## Caesium and rubidium gyromagnetic factors

Concerning caesium-133 and rubidium-87, the computation of the sensitivity has evolved over time. Using a chiral perturbation theory, it was first deduced (Flambaum et al. 2004) that

$$g_{\text{Cs}} \propto X_{\text{q}}^{0.110} X_{\text{s}}^{0.017} \propto X_{\text{q}}^{0.127}, \quad g_{\text{Rb}} \propto X_{\text{q}}^{-0.064} X_{\text{s}}^{-0.010} \propto X_{\text{q}}^{-0.074}.$$

where the second equality assumes $X_s \propto X_q$. Flambaum and Tedesco (2006) refined the computation thanks to three methods (see their Table III). Considering only the valence nucleon, they first got the same values listed just above as previously obtained by Flambaum et al. (2004) while including the non-valence nucleons lowered the coefficients respectively to 0.044 and -0.056. Further, including the effect of quark mass on the spin-spin interaction, they concluded

$$g_{\text{Cs}} \propto X_{\text{q}}^{0.009}, \quad g_{\text{Rb}} \propto X_{\text{q}}^{-0.016}. \qquad (103)$$

Taking into account the effect of the variation of the nuclear radius, it was corrected (Dinh et al. 2009) by a factor $X_q^{-0.007}$ and $X_q^{-0.003}$ respectively so that

$$g_{\text{Cs}} \propto X_{\text{q}}^{0.002}, \quad g_{\text{Rb}} \propto X_{\text{q}}^{-0.019}. \qquad (104)$$

One shall take these sensitivities with care. Luo et al. (2011) compared the computation of the dependencies of the gyromagnetic factor and proton mass to the light quark masses and shown that is strongly model-dependent. They first related the nuclear gyromagnetic factor to the ones of the proton and neutron as well as the spin-spin interaction b taking into account the effects of the polarization of the non-valence nucleons and spin-spin interaction

$$\frac{\delta g_{\text{Rb}}}{g_{\text{Rb}}} = 0.764 \frac{\delta g_{\text{p}}}{g_{\text{p}}} - 0.172 \frac{\delta g_{\text{n}}}{g_{\text{n}}} - 0.379 \frac{\delta b}{b}, \qquad (105)$$

$$\frac{\delta g_{\text{Cs}}}{g_{\text{Cs}}} = -0.619 \frac{\delta g_{\text{p}}}{g_{\text{p}}} + 0.152 \frac{\delta g_{\text{n}}}{g_{\text{n}}} + 0.335 \frac{\delta b}{b}. \qquad (106)$$

In comparison, the shell model gives

$$\frac{\delta g_{\text{Rb}}}{g_{\text{Rb}}} = 0.736 \frac{\delta g_{\text{p}}}{g_{\text{p}}}, \quad \frac{\delta g_{\text{Cs}}}{g_{\text{Cs}}} = -1.266 \frac{\delta g_{\text{p}}}{g_{\text{p}}}. \qquad (107)$$

The main difference arises from the dependence in $g_n$ and b but the order of magnitude is similar. The comparison of (1) the non-relativistic constituent quark model approach (2) chiral perturbation theory approach with and without combining with lattice QCD exhibited an important model-dependence in the computation of the gyromagnetic factors in terms of the quark masses and QCD scale (Luo et al. 2011).

## Influence of the s and c quark

Flambaum and Munro-Laylim (2023) computed the sensitivities of several secondary parameters to the s and c quark masses. It allowed them to conclude that
$\delta B_D/B_D \propto X_s^{-13}X_c^{-0.95}$ so that focusing on this single parameter, BBN implies that $|\delta m_s/m_s| < 1.8 \times 10^{-3}$ and $|\delta m_c/m_c| < 2.5 \times 10^{-2}$. Similarly, from the behavior of the resonance energy $E_r$ of $^{149}$Sm, they concluded from Oklo data that $|\delta m_s/m_s| < 1.7 \times 10^{-10}$ and $|\delta m_c/m_c| < 1.2 \times 10^{-9}$. Finally, from the scaling $\delta \bar{\mu}/\bar{\mu} \propto X_s^{0.056}X_c^{0.083}$, they concluded that atomic clock experiments set $|\dot{m}_s/m_s| < 7.9 \times 10^{-16} \text{ yr}^{-1}$ and $|\dot{m}_c/m_c| < 5.3 \times 10^{-16} \text{ yr}^{-1}$.

## Conclusions

Most of the data in the literature are interpreted in terms of the sensitivities computed by Flambaum and Tedesco (2006) and Dinh et al. (2009) that we adopt for the following numerical applications.

The sensitivities in $X_q$ for the hyperfine tandisyions of $^1\text{H}$, $^{133}\text{Cs}$ and $^{87}\text{Rb}$ arising from their gyromagnetic factors are taken to be

$$K_H^q = -0.1, \quad K_{Cs}^q = 0.002, \quad K_{Rb}^q = -0.019. \qquad (108)$$

When the dependence on $X_q$ arising from $m_e/m_p$ is included, Eq. (96) gives that

$$K^q \rightarrow K^q - 0.048. \qquad (109)$$

## 4.4 Expression of the atomic transitions

As detailed in Sect. 5.1.1, the combination of clocks allows one to set constraints on the variation of $(\alpha_{\text{EM}}, \bar{\mu})$ and the various gyromagnetic factors, more particularly (see Table 9) $g_p$, $g_{Cs}$ and $g_{Rb}$. The first step is a standard and straightforward analysis of the data to set constraints on the independent variations of the five primary QED parameters $(\alpha_{\text{EM}}, \bar{\mu}, g_p, g_{Cs}, g_{Rb})$.

Indeed, these parameters are not independent and extracting constraints on more fundamental parameters requires further theoretical insight. The first step, staying within QED, is to express the gyromagnetic factors in terms of $g_p$ and $g_n$, as described in Sect. 4.3. This can be performed in several ways to get either the sets of parameters $(\alpha_{\text{EM}}, \bar{\mu}, g_p, g_n, b)$ thanks to Eqs. (105-106) or simply $(\alpha_{\text{EM}}, \bar{\mu}, g_p)$ in the shell model (107). An example of such an analysis can be found in Luo et al. (2011), Ferreira et al. (2014). The next steps require the use of quantum chromodynamics, in particular to relate the nucleon $g$-factors in terms of the quark mass and the QCD scale as described in Eq. (101, 102) to express all the transition frequencies in terms of $(\alpha_{\text{EM}}, \bar{\mu}, X_q, X_s)$, recalling that the $X_{q,s}$ are defined in Eq. (3) with $m_q = (m_u + m_d)/2$. Indeed $\bar{\mu}$ includes the proton mass that depends of $X_{q,s}$ and $\alpha_{\text{EM}}$ so that one can shift to the parameters $(\alpha_{\text{EM}}, X_q, X_s, X_e)$ thanks to Eq. (94, 96).
<table><caption>**Table 6** Sensitivities of various transitions on a variation of the fine-structure constant, $\kappa_{\alpha}$ as defined in Eqs. (110) and (128)</caption><thead><tr><th>Atom</th><th>Transition</th><th>K<sub>α</sub></th><th>λ<sub>α</sub></th><th>λ<sub>μ</sub></th><th>λ<sub>κ<sub>α</sub></sub></th><th>K<sup>α</sup></th><th>K<sup>μ</sup></th><th>K<sup>q</sup></th><th>K<sup>e</sup></th></tr></thead><tbody><tr><td><sup>1</sup>H</td><td>1s &ndash; 2s</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>1</sup>H</td><td>hfs</td><td>0</td><td>2</td><td>1</td><td>2</td><td>1</td><td>-0.1</td><td>1</td><td>1</td></tr><tr><td><sup>133</sup>Cs</td><td>hfs</td><td>0.83</td><td>2.83</td><td>1</td><td>2.83</td><td>1</td><td>0.002</td><td>1</td><td>1</td></tr><tr><td><sup>87</sup>Rb</td><td>hfs</td><td>0.34</td><td>2.34</td><td>1</td><td>2.34</td><td>1</td><td>-0.019</td><td>1</td><td>1</td></tr><tr><td><sup>87</sup>Sr</td><td><sup>1</sup>S<sub>0</sub> &ndash; <sup>3</sup>P<sub>0</sub></td><td>0.06</td><td>0.06</td><td>0</td><td>0.06</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>27</sup>Al<sup>+</sup></td><td><sup>1</sup>S<sub>0</sub> &ndash; <sup>3</sup>P<sub>0</sub></td><td>0.008</td><td>0.008</td><td>0</td><td>0.008</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>199</sup>Hg<sup>+</sup></td><td><sup>2</sup>S<sub>1/2</sub> &ndash; <sup>2</sup>D<sub>5/2</sub></td><td>-2.94<sup>†</sup></td><td>-2.94</td><td>0</td><td>-2.94</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>171</sup>Yb<sup>+</sup></td><td>E2</td><td>1.0<sup>†</sup></td><td>1.0</td><td>0</td><td>0</td><td>1.0</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>171</sup>Yb<sup>+</sup></td><td>E3</td><td>-5.95</td><td>-5.95</td><td>0</td><td>0</td><td>-5.95</td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>162</sup>Dy</td><td>235 MHz</td><td>1.72 &times; 10<sup>7</sup></td><td>1.72 &times; 10<sup>7</sup></td><td>0</td><td>0</td><td>1.72 &times; 10<sup>7</sup></td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>163</sup>Dy</td><td>3.1 MHz</td><td>8.5 &times; 10<sup>6</sup></td><td>8.5 &times; 10<sup>6</sup></td><td>0</td><td>0</td><td>8.5 &times; 10<sup>6</sup></td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>164</sup>Dy</td><td>754 MHz</td><td>-2.6 &times; 10<sup>6</sup></td><td>-2.6 &times; 10<sup>6</sup></td><td>0</td><td>0</td><td>-2.6 &times; 10<sup>6</sup></td><td>0</td><td>0</td><td>0</td></tr><tr><td><sup>171</sup>Yb</td><td><sup>1</sup>S<sub>0</sub> &ndash; <sup>3</sup>P<sub>0</sub></td><td>0.31</td><td>0.31</td><td>0</td><td>0</td><td>0.31</td><td>0</td><td>0</td><td>0</td></tr><tr><td>SF<sub>6</sub></td><td>P(4)E<sub>0</sub></td><td>0</td><td>0</td><td>1/2</td><td>0</td><td>0</td><td>0.5</td><td>0</td><td>0.5</td></tr><tr><td>Krb</td><td></td><td>0</td><td>0</td><td>14890 &plusmn; 60</td><td>0</td><td>0</td><td>0.5</td><td>0</td><td>14890 &plusmn; 60</td></tr></tbody></table>

The coefficients $\lambda$ are defined in Eq. (110). The coefficients K are defined in Eq. (112) with $K^{\alpha} = \lambda_{\alpha}$ and $K^{\mu} = \lambda_{\mu}$. K$^{e}$ is defined in Eq. (114) in which the dependence of $m_{p}$ in $\alpha_{EM}$ is neglected so that $K^{\alpha} = \lambda_{\alpha}$. Notes, (‡) : the early value of this sensitivity was estimated as -3.2 (Dzuba et al. 1999a) and -3.19 (Peik et al. 2004) before reevaluated to -2.9 (Fortier et al. 2007) -2.94 (Flambaum and Dzuba 2009; Safronova 2019). (†) : the early value was 0.88 (Dzuba et al. 2003; Peik et al. 2008) then reevaluated to 1.03 (Safronova 2019) and 1.0 (Tamm et al. 2014)
## Definition of the physical interpretation schemes

Several interpretation schemes have been used in the literature with different model-dependent layers:

1. sharp and model independent constraints on the variation of the primary QED parameters ($\alpha_{EM}$, $\bar{\mu}$, $g_p$, $g_{Cs}$, $g_{Cs}$), using the sensitivity $\lambda$ summarized in Table 6 and defined as

$$v_A \propto g_A^{\lambda_g} \alpha_{EM}^{\lambda_x} \bar{\mu}^{\lambda_\mu} \qquad (110)$$

The sensitivity parameters are gathered in Table 6.

2. Expressing the gyromagnetic factors in terms of the quark masses using the expressions (101) and (104), one gets the frequencies in terms of ($\alpha_{EM}$, $\bar{\mu}$, $X_q$, $X_s$), as

$$v_A \propto \alpha_{EM}^{\lambda_x} \bar{\mu}^{\lambda_\mu} X_q^{K_q} X_s^{K_s}. \qquad (111)$$

Most of the constraints on atomic clocks assume $X_s \propto X_q$ so that it reduces to

$$v_A \propto \alpha_{EM}^{K_x} \bar{\mu}^{K_\mu} X_q^{K_q}, \qquad (112)$$

with $K_x = \lambda_x$ and $K_\mu = \lambda_\mu$.

3. The last step is to express the proton mass in $\bar{\mu}$. Neglecting the $\alpha_{EM}$-dependence of $m_p$, Flambaum et al. (2004) concluded the nucleon masses scale as Eq. (94). Hence, all transitions are expressed in terms of the fundamental parameters ($\alpha_{EM}$, $X_q$, $X_s$, $X_e$) as

$$v_A \propto \alpha_{EM}^{K_x} X_q^{K_q - 0.037} X_s^{K_s - 0.011} X_e \qquad (113)$$

and, in the approximation $X_q \propto X_s$,

$$v_A \propto \alpha_{EM}^{K_x} X_q^{K_q - 0.048 K_e} X_e^{K_e} \qquad (114)$$

and $K_e = 1$ for the transition with $K_\mu \neq 0$ and 0 otherwise. Note that the relation (94) assumes that $m_p \propto \Lambda_{QCD}$ so that $K_x = \kappa_x$, as assumed here. The values of the sensitivity coefficients are gathered in Table 6.

4. Sharper constraints under the hypothesis of correlated variations in unification schemes can also be set, e.g., in the (R, S)-formalism; see Sect. 4.5.

Indeed, these constraints are model-dependent and will depend on the physical modelizations required to compute the sensitivity parameters, as an example, Table III of Flambaum and Tedesco (2006) compares their values when different nuclear effects are considered. For instance, $K_q$ can vary from 0.127, 0.044 to 0.009 for the caesium according to whether one includes only valence nucleon, non-valence non-nucleon or effect of the quark mass on the spin-spin interaction; see also Luo et al. (2011) for a discussion of the model-dependence.

## Atomic clock data interpretations

The comparison of atomic clocks gives access to
<table><caption>**Table 7** Summary of the sensitivity parameters for clock comparison</caption><thead><tr><th>Clocks (A, B)</th><th>y<sub>AB</sub></th><th>ΔK<sup>α</sup></th><th>ΔK<sup>μ</sup></th><th>ΔK<sup>q</sup></th><th>ΔK<sup>e</sup></th></tr></thead><tbody><tr><td>Rb-Cs</td><td>(g<sub>Rb</sub>/g<sub>Cs</sub>)α<sub>EM</sub><sup>-0.49</sup></td><td>-0.49</td><td>0</td><td>-0.021</td><td>0</td></tr><tr><td>H(hfs)-Cs</td><td>(g<sub>p</sub>/g<sub>Cs</sub>)α<sub>EM</sub><sup>-0.83</sup></td><td>-0.83</td><td>0</td><td>-0.102</td><td>0</td></tr><tr><td>H(1 s-2 s)-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>2.83</sup></td><td>-2.83</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>Hg<sup>+</sup>-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>5.77</sup></td><td>-5.77</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>Sr-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>2.77</sup></td><td>-2.77</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>Yb(<sup>3</sup>P<sub>0</sub>)-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>2.52</sup></td><td>-2.52</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>Yb(E2)-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>1.83</sup></td><td>-1.83</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>Yb(E3)-Cs</td><td>(g<sub>Cs</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>8.78</sup></td><td>-8.78</td><td>-1</td><td>-0.002</td><td>-1</td></tr><tr><td>CSO-H(hfs)</td><td>(g<sub>p</sub>μ)<sup>-1</sup>α<sub>EM</sub><sup>3</sup></td><td>3</td><td>-1</td><td>0.1</td><td>-1</td></tr><tr><td>Hg<sup>+</sup>-Al<sup>+</sup></td><td>α<sub>EM</sub><sup>2.948</sup></td><td>-2.948</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Yb(E3)-Yb(E2)</td><td>α<sub>EM</sub><sup>6.95</sup></td><td>6.95</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Dy</td><td>α<sub>EM</sub></td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>SF<sub>6</sub>-Cs</td><td>g<sub>Cs</sub><sup>-1</sup>μ<sup>-1/2</sup>α<sub>EM</sub><sup>2.83</sup></td><td>-2.83</td><td>-0.5</td><td>-0.002</td><td>-0.5</td></tr><tr><td>KRb-Cs</td><td>g<sub>Cs</sub><sup>-1</sup>μ<sup>14890±60</sup>α<sub>EM</sub><sup>-2.83</sup></td><td>-2.83</td><td>14890 ± 60</td><td>-0.002</td><td>14890 ± 60</td></tr></tbody></table>

The first column indicates the clock comparisons, the second gives the expression of y<sub>AB</sub> as defined in Eq. (115) in terms of (α<sub>EM</sub>, μ, α<sub>A</sub>) thanks to Eq. (110). The columns 3 to 6 give the coefficients ΔK<sub>AB</sub> defined in Eq. (118) and derived from the values listed in Table 6. Note that the value of ΔK<sup>q</sup> in column 5 corresponds to the decomposition (112) in terms of (α<sub>EM</sub>, μ, X<sub>q</sub>) and that one needs to take into account the shift (109) for ΔK<sup>q</sup> when using the parameters (α<sub>EM</sub>, X<sub>q</sub>, X<sub>e</sub>); see Eq. (114)

$$ \delta \ln y_{AB} \equiv \delta \ln \frac{v_A}{v_B} \qquad (115) $$

that can be expressed as

$$ \delta \ln y_{AB} = \Delta \lambda_{\alpha} \frac{\delta \alpha_{EM}}{\alpha_{EM}} + \Delta \lambda_{\mu} \frac{\delta \bar{\mu}}{\bar{\mu}} + \Delta \lambda_{g_i} \frac{\delta g_i}{g_i} \qquad (116) $$

with the sensitivity coefficients summarized in Table 7 or as

$$ \delta \ln y_{AB} = \Delta K_{\alpha} \frac{\delta \alpha_{EM}}{\alpha_{EM}} + \Delta K_{\mu} \frac{\delta \bar{\mu}}{\bar{\mu}} + \Delta K_{g_i} \frac{\delta X_{q_i}}{X_{q_i}} \qquad (117) $$

with the coefficients

$$ \Delta_{AB} K^i = K_A^i - K_B^i \qquad (118) $$

gathered in Table 7.

### Astrophysical data interpretation

Similarly, as the previous analysis for the constraints obtained on QSO observables defined and discussed in Sects. 5.6.1 to 5.7 can be expressed as
$$
\begin{align}
y &\equiv g_p \alpha_{\text{EM}}^2 \nonumber \\
\bar{\mu} &\equiv \frac{m_e}{m_p} \nonumber \\
x &\equiv g_p \alpha_{\text{EM}}^2 \mu \nonumber \\
F &\equiv g_p (\alpha_{\text{EM}}^2 \mu)^{1.57} \nonumber \\
F' &\equiv \alpha_{\text{EM}}^2 \mu \nonumber \\
G &\equiv g_p (\alpha_{\text{EM}}^2 \mu)^{1.85} \tag{119}
\end{align}
$$

once the scaling of the nucleon mass (94) is used, so that the seven observable quantities can be reduced to only 4 parameters.

4.5 Phenomenological (R, S)-parameterisation

As seen from the previous paragraph, the simplest self-consistent way to
phenomenologically describe models which correlated variations is to relate them
to α<sub>EM</sub>. A broad class of grand unification models have been encompassed in the
(R, S) parameterisation.

Definition

Following Campbell and Olive (1995), these two parameters are defined by
Eqs. (88) and (87) respectively which implies

$$
\Delta v / v = S \Delta h / h, \quad \Delta \Lambda_{QCD} / \Lambda_{QCD} = R \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}.
$$

Assuming all Yukawa couplings enjoy a universal variation, one deduces that

$$
\Delta m_e / m_e = \frac{1}{2} (1 + S) \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}, \quad (120)
$$

and, within a dilaton model, that allows a relation between $\Delta h/h$ and $\Delta \alpha_{\text{EM}}/\alpha_{\text{EM}}$, one has

$$
\Delta m_n / m_n = \Delta m_p / m_p = [0.8R + 0.2(1+S)] \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}. \quad (121)
$$

The relevant BBN parameters can then be shown to behave as (Coc et al. 2007)

$$
\Delta Q_{np} / Q_{np} = [0.1 + 0.7S - 0.6R] \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}, \quad (122)
$$

$$
\Delta \tau_n / \tau_n = [-0.2 - 2.0S + 3.8R] \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}, \quad (123)
$$

$$
\Delta B_D / B_D = [-6.5(1+S) + 18R] \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}, \quad (124)
$$

while for clock studies, one has (Martins and Pinho 2017)

$$
\Delta \bar{\mu} / \bar{\mu} = P \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}}, \quad \Delta g_p / g_p = Q \Delta \alpha_{\text{EM}} / \alpha_{\text{EM}} \quad (125)
$$

with $R = 10(2P - 15Q)$ and $(1+S) = 50(P - 8Q)$.
<figure><img src="image_18.png" /><figcaption>**Fig. 11** Summary of the systems that have been used to probe the constancy of the fundamental constants and their position in a spacetime diagram in which the cone represents our past light cone. The shaded areas represents the comoving space probed by different tests. The scale of the left indicates the typical order of magnitude of the constraints on $\Delta\alpha_{EM}/\alpha_{EM}$; e.g., -6 means $\Delta\alpha_{EM}/\alpha_{EM} \lesssim 10^{-6}$ at the exception of the first one which refers to $\alpha_{EM}/\alpha_{EM} \lesssim 10^{-19}/\text{yr}$. We also indicate the typical redshift $z$ of the system</figcaption></figure>

$R$ and $S$ can be taken as pure phenomenological parameters. One expects $S > 0$ while $R$ can be any sign. Their typical value in unification scenarios are thought to be $(R, S) = (26, 160)$ (Langacker et al. 2002; Coc et al. 2007) and (109.4,0) in dilaton type models (Nakashima et al. 2010).

### Constraints on (R, S)

These two parameters can be constrained by observation. Many works have interpreted data within this framework (see discussion in Sect. 4.1). Luo et al. (2011) analyzed atomic clocks constraints (see Sect. 5.1.1) in this framework as well as Julião et al. (2014) that showed that the data are degenerated according to $(S+1) - 2.7R = -5 \pm 15$ and that $P = 1.5 \pm 4.5$. Ferreira et al. (2013, 2014), Martins and Pinho (2017) stressed that the various observation of PKS1413+135 (see Table 14) allows one to set independent constraints on $(\alpha_{EM}, \mu, g_p)$ and then extracted from the QSO constraints the one-dimensional confidence intervals for $(R, S) = (277 \pm 24, 742 \pm 65)$ and from molecular data within our Galaxy (João et al. 2015), while Thompson (2017a, 2017b) interpreted molecular spectra. Martins and Vila Miñana (2019) combined local and astrophysical data. From a model of polytropic white dwarfs with varying constants, Magano et al. (2017) determined their mass-radius relation dependence on (R, S). From G191-B2B (see Sect. 8.2.4) it was concluded that it constraints a different combination than atomic clocks.
<table><caption>**Table 8** Summary of the systems considered to set constraints on the variation of the fundamental constants</caption><thead><tr><th>System</th><th>Observable</th><th>Primary constraints</th><th>Other hypothesis</th><th>Section</th></tr></thead><tbody><tr><td>Atomic clock</td><td>δ ln v</td><td>g<sub>i</sub>, α<sub>EM</sub>, μ</td><td>–</td><td>5.1</td></tr><tr><td>Oklo phenomenon</td><td>isotopic ratio</td><td>E<sub>r</sub></td><td>geophysical model</td><td>5.3</td></tr><tr><td>Meteorite dating</td><td>isotopic ratio</td><td>λ</td><td>–</td><td>5.4</td></tr><tr><td>Quasar spectra</td><td>atomic spectra</td><td>g<sub>p</sub>, μ, α<sub>EM</sub></td><td>cloud physical properties</td><td>5.5</td></tr><tr><td>Stellar physics</td><td>element abundances</td><td>B<sub>D</sub></td><td>stellar model</td><td>5.9</td></tr><tr><td>CMB</td><td>ΔT/T</td><td>μ, α<sub>EM</sub></td><td>cosmological model</td><td>5.10</td></tr><tr><td>21 cm</td><td>T<sub>b</sub>/T<sub>CMB</sub></td><td>g<sub>p</sub>, μ, α<sub>EM</sub></td><td>cosmological model</td><td>5.11</td></tr><tr><td>Galaxy clusters</td><td>SZ effect</td><td>y</td><td>galaxy+cosmological model</td><td>5.12</td></tr><tr><td>BBN</td><td>light element abundances</td><td>Q<sub>np</sub>, τ<sub>n</sub>, m<sub>e</sub>, m<sub>N</sub>, α<sub>EM</sub>, B<sub>D</sub></td><td>cosmological model</td><td>5.14</td></tr></tbody></table>

We summarize the observable quantities, the primary constants used to interpret the data and the other hypothesis required for their interpretation. All the quantities appearing in this table are defined in the text

## 5 Experimental and observational constraints on non-gravitational constants

This section focuses on the experimental and observational constraints on the non-gravitational constants, that is assuming $α_G$ remains constant. We use the convention that $Δα = α - α_0$ for any constant $α$, so that $Δα < 0$ refers to a value smaller in the past than today.

### Physical systems

The various physical systems that have been considered to constrain the variation of the fundamental constants can be classified in many ways. We can classify them according to chronologically thanks to their look-back time and more precisely by their spacetime position relative to our actual position. This is summarized in Fig. 11. Indeed higher redshift systems offer the possibility to set constraints on larger time scales, but this is at the expense of usually involving other parameters such as the cosmological parameters. This is, in particular, the case of the cosmic microwave background or of primordial nucleosynthesis. The systems can also be classified in terms of the physics they involve. For instance, atomics clocks, quasar absorption spectra and the cosmic microwave background require only to use quantum electrodynamics to draw the primary constraints while the Oklo phenomenon, meteorites dating and nucleosynthesis require nuclear physics. This is summarized on Table 8.
<figure><img src="image_19.png" /><figcaption>**Fig. 12** Given a physical system, its modelization will allow one to (1) compute the sensitivity of the observables *O* to the relevant list of primary parameters *G*<sub>*k*</sub> by computing the sensitivity coefficients (441) and (2) identifying all the external parameters *X* to be controlled such as e.g., the temperature, magnetic fields etc... This allows one to draw constraints on the independent variation of the primary parameters. Then those parameters need to be expressed in terms of the fundamental constants α<sub>*i*</sub> thanks to the sensitivity coefficients (442). One can then investigate models with correlated or uncorrelated variations to draw final conclusions on the correlated or independent variations of the fundamental constants. The definitions of the sensitivity coefficients are summarized in Appendix B.2</figcaption></figure>

## Strategy

For any system, setting constraints goes through several steps, summarized on Fig. 12.

* First, we have some observable quantities *O* from which we can draw constraints on primary parameters *G*<sub>*k*</sub>, which may or may not be fundamental constants (e.g., the BBN parameters, the lifetime of β-decayers, ...). This allows us to state which constants dominate the system and which can be neglected in the analysis. Given data and a study of the effect of the external physical parameters of the system, one can draw constraints on the independent variations of the *G*<sub>*k*</sub>.

* These primary parameters must then be related to a set of fundamental constants. This requires e.g., to connect nuclear physics to QCD or atomic physics to QED. One can then get constraints on the independent variations of the fundamental constants α<sub>*i*</sub> as described in Sect. 4.2.

* In a last step, the number of constants can be reduced by relating them in some unification schemes to get stronger constraints at the expense of being more model-depedent, as described in Sect. 4.1.
Indeed each step requires a specific modelization and hypothesis and has its own limitations.

## 5.1 Atomic and molecular clocks

The search of the variation of fundamental constants with clocks has followed the developments and progresses of metrology and in particular the tremendous developments of atomic clock precision in the past 20 years; see e.g., the reviews by Safronova et al. (2018a), Safronova (2019). While most of the constraints so far have been established with atomic clocks, many new techniques including highly charged ion, molecular and nuclear clocks are now considered.

### 5.1.1 Atomic spectra and constants

The laboratory constraints on the time variation of fundamental constants are obtained by comparing the long-term behavior of several oscillators and rely on frequency measurements. They have witnessed tremendous progresses over the past years, in particular due to the central role of time and frequency measurements in metrology.

#### *Generalities*

The atomic transitions have various dependencies in the fundamental constants that can be used in clocks comparison to constrain their variation. As a textbook example, for the hydrogen atom, the gross, fine and hyperfine-structures are roughly given by

$$2p - 1s : v \propto cR_{\infty}, \quad 2p_{3/2} - 2p_{1/2} : v \propto cR_{\infty}\alpha_{\text{EM}}^2, \quad 1s : \propto cR_{\infty}\alpha_{\text{EM}}^2 g_{\text{p}}\bar{\mu},$$

respectively, where the Rydberg constant set the dimension. $g_{\text{p}}$ is the proton gyromagnetic factor and $\bar{\mu} = m_e/m_{\text{p}}$.

In the non-relativistic approximation, the transitions of all atoms have similar dependencies but two effects have to be taken into account.

* First, the hyperfine-structures involve a gyromagnetic factor $g_i$ (related to the nuclear magnetic moment by $\mu_i = g_i\mu_N$, with $\mu_N = e\hbar/2m_{\text{p}}c$), which are different for each nuclei.

* Second, relativistic corrections (including the Casimir contribution), which also depend on each atom (but also on the type of the transition) can be included through a multiplicative function $F_{\text{rel}}(\alpha_{\text{EM}})$. It has a strong dependence on the atomic number Z, which can be illustrated on the case of alkali atoms, for which

$$F_{\text{rel}}(\alpha_{\text{EM}}) = [1 - (Z\alpha_{\text{EM}})^2]^{-1/2} \left[1 - \frac{4}{3}(Z\alpha_{\text{EM}})^2\right]^{-1} \approx 1 + \frac{11}{6}(Z\alpha_{\text{EM}})^2.$$
The developments of highly accurate atomic clocks using different transitions in different atoms offer the possibility to test various combinations of the fundamental constants.

## Dependencies on the fundamental constants

It follows that at the lowest level of description, we can interpret all atomic clocks results in terms of the g-factors of each atoms, $g_i$, the electron to proton mass ration $\bar{\mu}$ and the fine-structure constant $\alpha_{EM}$. We shall parameterize the hyperfine and fine-structures frequencies as follows.

* The hyperfine frequency in a given electronic state of an alkali-like atom, such as $^{133}$Cs, $^{87}$Rb, $^{199}$Hg$^+$, is

$$v_{\text{hfs}} \approx R_{\infty}c \times A_{\text{hfs}} \times g_i \times \alpha_{\text{EM}}^2 \times \bar{\mu} \times F_{\text{hfs}}(\alpha_{\text{EM}}) \quad (126)$$

where $g_i = \mu_i/\mu_N$ is the nuclear g-factor. $A_{hfs}$ is a numerical factor depending on each particular atom and we have set $F_{rel} = F_{hfs}(\alpha)$.

* The frequency of an electronic transition is well-approximated by

$$v_{\text{elec}} \approx R_{\infty}c \times A_{\text{elec}} \times F_{\text{elec}}(Z, \alpha), \quad (127)$$

where, as above, $A_{elec}$ is a numerical factor depending on each particular atom and $F_{elec}$ is the function accounting for relativistic effects, spin-orbit couplings and many-body effects. Even though an electronic transition should also include a contribution from the hyperfine interaction, it is generally only a small fraction of the transition energy and thus should not carry any significant sensitivity to a variation of the fundamental constants.

## Sensitivity parameters

The importance of the relativistic corrections was probably first emphasized in Pretage et al. (1995) and their computation through relativistic N-body calculations was carried out for many transitions in Dzuba and Flambaum (2001), Dzuba et al. (1999a, 2003), Flambaum (2004). They can be characterized by introducing the sensitivity of the relativistic factors to a variation of $\alpha_{EM}$,

$$\kappa_{\alpha} \equiv \frac{\partial \ln F}{\partial \ln \alpha_{\text{EM}}}. \quad (128)$$

Table 6 summarizes the values of some of them, as computed in Dzuba et al. (1999a), Flambaum and Tedesco (2006). Indeed a reliable knowledge of these coefficients at the 1% to 10% level is required. The interpretation of the spectra in this context relies, from a theoretical point of view, only on quantum electrodynamics (QED), a theory, which is well tested experimentally (Karshenboim 2005c) so that we can safely obtain constraints on ($\alpha_{EM}, \bar{\mu}, g_i$), still keeping in mind that the computation of the sensitivity factors required numerical N-body simulations. In terms of the primary parameters ($\alpha_{EM}, \bar{\mu}, g_A$), any transitions takes the general
form (110), $v_A \propto cR_\infty g_A^{\lambda_g} \alpha_{EM}^{\lambda_x} \bar{\mu}^{\lambda_\mu}$ where the sensitivity coefficients $\lambda_i$ are gathered in Table 6.

This implies that the comparison of two clocks allows one to constrain different combinations of constants since (1) the comparison of two microwave clock frequencies depends on the gyromagnetic factors and $\alpha_{EM}$ but not on $\bar{\mu}$, (2) the comparison of two optical clocks depends on $\alpha_{EM}$ only and (3) the comparison of an optical to microwave clocks depends on $(\alpha_{EM}, \bar{\mu}, g_A)$.

### 5.1.2 Experimental constraints from atomic clocks

We present the latest results that have been obtained from the comparison of atomic clocks and refer to Sect. III.B.2 of FCV (Uzan 2003b) for earlier studies. They all rely on the developments of new atomic clocks, with the primarily goal to define better frequency standards; see Dimarcq et al. (2024). From an experimental point of view, various combinations of clocks have been performed. It is important to analyze as many species as possible in order to rule-out species-dependent systematic effects. Most experiments are based on a frequency comparison to caesium clocks. The hyperfine splitting frequency between the $F=3$ and $F=4$ levels of its $^2S_{1/2}$ ground state at 9.192 GHz has been used for the definition of the second since 1967. One limiting effect, that contributes mostly to the systematic uncertainty, is the frequency shift due to cold collisions between the atoms. On this particular point, clocks based on the hyperfine frequency of the ground state of the rubidium at 6.835 GHz, are more favorable.

In order to be as independent as possible from theoretical hypothesis, this section presents only “raw” constraints on clock comparisons.

#### Atomic hydrogen vs caesium-133

The $1s-2s$ transition in atomic hydrogen was compared to the ground state hyperfine splitting of caesium-133 (Fischer et al. 2004) in 1999 and 2003, setting an upper limit on the variation of $v_H$ of $(-29 \pm 57)$ Hz within 44 months. This can be translated in a relative drift

$$ \frac{d}{dt} \ln \left( \frac{v_H}{v_{Cs}} \right) = (-32 \pm 63) \times 10^{-16} \text{yr}^{-1}. \qquad (129) $$

Since the relativistic correction for the atomic hydrogen transition nearly vanishes, we have $v_H \sim R_\infty$ so that

$$ \frac{v_H}{v_{Cs}} \propto (g_{Cs} \bar{\mu})^{-1} \alpha_{EM}^{-2.83}. $$

#### Rubidium-87 vs caesium-133

The comparison of the ground state hyperfine frequencies of the rubidium-87 and caesium-133 in laser-cooled atomic fountain clocks in their electronic ground state between 1998 and 2003, with an accuracy of order $10^{-15}$ allowed Marion et al. (2003) to get $\frac{d}{dt} \ln \left( \frac{v_{Rb}}{v_{Cs}} \right) = (0.2 \pm 7.0) \times 10^{-16} \text{yr}^{-1}$. With one more year of
<table><caption>**Table 9** Summary of the experimental constraints obtained from the comparisons of atomic clocks</caption><thead><tr><th>Clock 1</th><th>Clock 2</th><th>Constraint (yr<sup>-1</sup>) d/dt ln(y<sub>clock1</sub>/y<sub>clock2</sub>)</th><th>Constants dependence</th><th>Reference</th></tr></thead><tbody><tr><td rowspan="4"><sup>87</sup>Rb</td><td rowspan="4"><sup>133</sup>Cs</td><td>(0.2 ± 7.0) × 10<sup>-16</sup></td><td rowspan="4"><sup>gRb</sup>/<sub>gCs</sub> α<sub>EM</sub><sup>-0.49</sup></td><td>Marion et al. (2003)</td></tr><tr><td>(-0.5 ± 5.3) × 10<sup>-16</sup></td><td>Bize et al. (2005)</td></tr><tr><td>(-1.36 ± 0.91) × 10<sup>-16</sup></td><td>Guena et al. (2012)</td></tr><tr><td>(-1.16 ± 0.61) × 10<sup>-16</sup></td><td>Abgrall et al. (2015)</td></tr><tr><td><sup>1</sup>H</td><td><sup>133</sup>Cs</td><td>(-32 ± 63) × 10<sup>-16</sup></td><td>(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-2.83</sup></td><td>Fischer et al. (2004)</td></tr><tr><td rowspan="2"><sup>199</sup>Hg<sup>+</sup></td><td rowspan="2"><sup>133</sup>Cs</td><td>(0.2 ± 7) × 10<sup>-15</sup></td><td rowspan="2">(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-5.77</sup></td><td>Bize et al. (2003)</td></tr><tr><td>(3.7 ± 3.9) × 10<sup>-16</sup></td><td>Fortier et al. (2007)</td></tr><tr><td rowspan="2"><sup>171</sup>Yb<sup>+</sup> (E2)</td><td rowspan="2"><sup>133</sup>Cs</td><td>(-0.78 ± 1.40) × 10<sup>-15</sup></td><td rowspan="2">(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-1.83</sup></td><td>Peik et al. (2008)</td></tr><tr><td>(-0.5 ± 1.9) × 10<sup>-16</sup></td><td>Tamm et al. (2014)</td></tr><tr><td rowspan="2"><sup>171</sup>Yb<sup>+</sup> (E3)</td><td rowspan="2"><sup>133</sup>Cs</td><td>(-0.2 ± 4.1) × 10<sup>-16</sup></td><td rowspan="2">(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-8.78</sup></td><td>Huntemann et al. (2014)</td></tr><tr><td>(-3.1 ± 3.4) × 10<sup>-17</sup></td><td>Lange et al. (2021)</td></tr><tr><td><sup>171</sup>Yb <sup>1</sup>S<sub>0</sub> <sup>-3</sup>P<sub>0</sub></td><td><sup>133</sup>Cs</td><td>(-4.9 ± 3.6) × 10<sup>-17</sup></td><td>(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-2.52</sup></td><td>McGrew et al. (2019a)</td></tr><tr><td rowspan="4"><sup>87</sup>Sr</td><td rowspan="4"><sup>133</sup>Cs</td><td>(-1.0 ± 1.8) × 10<sup>-15</sup></td><td rowspan="4">(g<sub>Cs</sub>μ)<sup>-1</sup> α<sub>EM</sub><sup>-2.77</sup></td><td>Blatt et al. (2008)</td></tr><tr><td>(-3.3 ± 3.0) × 10<sup>-16</sup></td><td>Le Targat et al. (2013)</td></tr><tr><td>(-2.3 ± 1.8) × 10<sup>-16</sup></td><td>Abgrall et al. (2015)</td></tr><tr><td>(-4.2 ± 3.3) × 10<sup>-17</sup></td><td>Schwarz et al. (2020)</td></tr><tr><td><sup>199</sup>Hg<sup>+</sup></td><td><sup>27</sup>Al<sup>+</sup></td><td>(5.3 ± 7.9) × 10<sup>-17</sup></td><td>α<sub>EM</sub><sup>-2.948</sup></td><td>Rosenband et al. (2008)</td></tr><tr><td rowspan="2"><sup>171</sup>Yb<sup>+</sup> (E3)</td><td rowspan="2"><sup>171</sup>Yb<sup>+</sup> (E2)</td><td>(-6.8 ± 7.8) × 10<sup>-18</sup></td><td rowspan="2">α<sub>EM</sub><sup>-6.95</sup></td><td>Lange et al. (2021)</td></tr><tr><td>(-1.2 ± 1.8) × 10<sup>-18</sup></td><td>Filzinger et al. (2023)</td></tr><tr><td><sup>162</sup>Dy</td><td><sup>163</sup>Dy</td><td>(-2.7 ± 2.6) × 10<sup>-15</sup></td><td>α<sub>EM</sub></td><td>Cingöz et al. (2008)</td></tr><tr><td><sup>162</sup>Dy</td><td><sup>164</sup>Dy</td><td>(-5.8 ± 6.9) × 10<sup>-17</sup></td><td>α<sub>EM</sub></td><td>Leefer et al. (2013)</td></tr><tr><td>SF<sup>6</sup></td><td><sup>133</sup>Cs</td><td>(1.9 ± 2.7) × 10<sup>-14</sup></td><td>g<sub>Cs</sub>μ<sup>-1/2</sup> α<sub>EM</sub><sup>-2.83</sup></td><td>Shelnikov et al. (2008)</td></tr></tbody></table>
<table><caption>**Table 9 continued**</caption><thead><tr><th>Clock 1</th><th>Clock 2</th><th>Constraint (yr<sup>-1</sup>)</th><th>d/dt ln(v<sub>clock1</sub>/v<sub>clock2</sub>)</th><th>Constants dependence</th><th>Reference</th></tr></thead><tbody><tr><td>KRb</td><td><sup>133</sup>Cs</td><td>(-0.44 ± 1.47<sub>stat</sub> ± 0.24<sub>syst</sub>) × 10<sup>-10</sup></td><td>g<sub>Cs</sub><sup>-1</sup>μ<sup>15 000</sup>α<sub>EM</sub><sup>-2</sup></td><td></td><td>Kobayashi et al. (2019)</td></tr></tbody></table>

For each constraint on the relative drift of the frequency ratio of the two clocks, we provide the dependence in the various constants, using the numbers of Table 6. The experiments on SF₆ and KRb molecular clocks are described in Sect. 5.1.5 below. Note that the constraints obtained from the same clock comparison are not independent (since they report results from the same experiment, running over different time spans); see text for discussion

experiment, the constraint dropped $\frac{d}{dt} \ln(v_{Rb}/v_{Cs}) = (-0.5 \pm 5.3) \times 10^{-16} \text{yr}^{-1}$. Guena et al. (2012) then reported the constraint from measurements spanning 14 years of experiment $\frac{d}{dt} \ln(v_{Rb}/v_{Cs}) = (-1.36 \pm 0.91) \times 10^{-16} \text{yr}^{-1}$. It was further improved by taking into account 3 more years of data by Abgrall et al. (2015) to

$$ \frac{d}{dt} \ln \left( \frac{v_{Rb}}{v_{Cs}} \right) = (-1.16 \pm 0.61) \times 10^{-16} \text{yr}^{-1}. \qquad (130) $$

From Eq. (126), and using the values of the sensitivities $\kappa_\alpha$, listed in Table 6, one gets

$$ \frac{v_{Rb}}{v_{Cs}} \propto \frac{g_{Rb}}{g_{Cs}} \alpha_{EM}^{-0.49}, $$

that has the advantage to be independent of $\bar{\mu}$.

### Aluminium and mercury single-ion optical clocks

The comparison of the $^1S_0 - ^3P_0$ transition in $^{27}$Al⁺ and $^2S_{1/2} - ^2D_{5/2}$ in $^{199}$Hg⁺ over a year allowed Rosenband et al. (2008) to set the constraint

$$ \frac{d}{dt} \ln \left( \frac{v_{Hg}}{v_{Al}} \right) = (5.3 \pm 7.9) \times 10^{-17} \text{yr}^{-1}. \qquad (131) $$

Proceeding as previously, this tests the stability of<fnref n="8" />

$$ \frac{v_{Hg}}{v_{Al}} \propto \alpha_{EM}^{-2.948}, $$

hence giving a sharp constraint on $\alpha_{EM}$ alone. Improvement of one order of magnitude are foreseen from the development of $^{27}$Al⁺ quantum-logic clock with a systematic uncertainty below $10^{-18}$ by Brewer et al. (2019).

<footnote n="8">⁸ Note that this sensitivity was estimated as -3.02/-3.208 and then reevaluated to -2.948; see Table 6.</footnote>
## Mercury-199 vs caesium-133

The ¹⁹⁹Hg⁺ ²S₁/₂ –²D₅/₂ optical transition has a high sensitivity to α<sub>EM</sub> so that it is well suited to test its variation. While ν<sub>Cs</sub> is still given by Eq. (126), ν<sub>Hg</sub> is given by Eq. (127). Using the sensitivities of Table 6, we conclude that

$$ \frac{v_{\text{Hg}}}{v_{\text{Cs}}} \propto (g_{\text{Cs}}\bar{\mu})^{-1} \alpha_{\text{EM}}^{-5.77}. $$

The frequency of the ¹⁹⁹Hg⁺ electric quadrupole transition at 282 nm was compared to the ground state hyperfine transition of caesium during a two year period, which lead to (Bize et al. 2003) $\frac{d}{dt} \ln(v_{\text{Hg}}/v_{\text{Cs}}) = (0.2 \pm 7) \times 10^{-15} \text{yr}^{-1}$. This was improved by a comparison over a 6 year period (Fortier et al. 2007) to get

$$ \frac{d}{dt} \ln \left( \frac{v_{\text{Hg}}}{v_{\text{Cs}}} \right) = (3.7 \pm 3.9) \times 10^{-16} \text{yr}^{-1}. \qquad (132) $$

## Strontium-87 vs caesium-133

The comparison of the ¹S₀ –³P₀ transition in neutral ⁸⁷Sr with a caesium clock was performed in three independent laboratories. The combination of these three experiments (Blatt et al. 2008) led to the constraint $\frac{d}{dt} \ln(v_{\text{Sr}}/v_{\text{Cs}}) = (-1.0 \pm 1.8) \times 10^{-15} \text{yr}^{-1}$.

Using a system of 2 Sr-clocks and 3-Cs clocks Le Targat et al. (2013) obtained $\frac{d}{dt} \ln(v_{\text{Sr}}/v_{\text{Cs}}) = (-3.3 \pm 3.0) \times 10^{-16} \text{yr}^{-1}$, later improved by Abgrall et al. (2015) to $\frac{d}{dt} \ln(v_{\text{Sr}}/v_{\text{Cs}}) = (-2.3 \pm 1.8) \times 10^{-16} \text{yr}^{-1}$. From a series of 42 measurements of the transition frequency of the ¹S₀–³P₀ line in ⁸⁷Sr against 2 caesium clocks at PTB between 2017 and 2019, Schwarz et al. (2020) concluded that

$$ \frac{d}{dt} \ln \left( \frac{v_{\text{Sr}}}{v_{\text{Cs}}} \right) = (-4.2 \pm 3.3) \times 10^{-17} \text{yr}^{-1}. \qquad (133) $$

Proceeding as previously, this tests the stability of

$$ \frac{v_{\text{Sr}}}{v_{\text{Cs}}} \propto (g_{\text{Cs}}\bar{\mu})^{-1} \alpha_{\text{EM}}^{-2.77}. $$

## Ytterbium

It is the only case among the clocks presently under development for which there is more than one clock transition. Yb⁺ has two ultranarrow optical clock transitions, an electric octupole (E3: ²S₁/₂–²F₇/₂) at 467 nm and an electric quadrupole (E2: ²S₁/₂ –²D₃/₂) at 436 nm (688 THz). This leads to various constraints.
* **Yb⁺-E2 vs caesium-133.** The E2 electric quadrupole transition was compared to the ground state hyperfine transition of caesium. The constraint of Peik et al. (2004) was updated, after comparison over a six year period, which led to (Peik et al. 2008; Peik 2010) $\frac{d}{dt} \ln(\frac{\nu_{Yb-E2}}{\nu_{Cs}}) = (-0.78 \pm 1.40) \times 10^{-15} \text{yr}^{-1}$. This was improved by Tamm et al. (2014) to get

$$ \frac{d}{dt} \ln \left( \frac{\nu_{Yb-E2}}{\nu_{Cs}} \right) = (-0.5 \pm 1.9) \times 10^{-16} \text{yr}^{-1}. \qquad (134) $$

Proceeding as previously, this tests the stability of

$$ \frac{\nu_{Yb-E2}}{\nu_{Cs}} \propto (g_{Cs}\bar{\mu})^{-1} \alpha_{EM}^{-1.83}. $$

* **Yb⁺-E3 vs caesium-133.** A similar analysis for the transition E3 at 467 nm gave (Huntemann et al. 2014) $\frac{d}{dt} \ln(\frac{\nu_{Yb-E3}}{\nu_{Cs}}) = (0.2 \pm 4.1) \times 10^{-16} \text{yr}^{-1}$. Besides, the comparison with a caesium clock allow Lange et al. (2021) to conclude that

$$ \frac{d}{dt} \ln \left( \frac{\nu_{Yb-E3}}{\nu_{Cs}} \right) = (-3.1 \pm 3.4) \times 10^{-17} \text{yr}^{-1}. \qquad (135) $$

Table 6 gives that

$$ \frac{\nu_{Yb-E3}}{\nu_{Cs}} \propto (g_{Cs}\bar{\mu})^{-1} \alpha_{EM}^{-8.78}. $$

* **Yb⁺-E3 vs Yb⁺-E2.** Godun et al. (2014) performed the first measurement of the ratio between E2 and E3 without reference to the Cs primary standard, and using the same single ion of ¹⁷¹Yb. Such a direct measurement of the ratio of the two optical frequencies are free from the additional uncertainties introduced by the primary Cs frequency standard. This was then used by Lange et al. (2021) who got

$$ \frac{d}{dt} \ln \left( \frac{\nu_{Yb-E3}}{\nu_{Yb-E2}} \right) = (-6.8 \pm 7.6) \times 10^{-18} \text{yr}^{-1}. \qquad (136) $$

The analysis of about 235 days of measurement data accumulated over a period of about 26 months, led Filzinger et al. (2023) to conclude that

$$ \frac{d}{dt} \ln \left( \frac{\nu_{Yb-E3}}{\nu_{Yb-E2}} \right) = (-1.2 \pm 1.8) \times 10^{-18} \text{yr}^{-1} \qquad (137) $$

once combined with the previous data by Lange et al. (2021). For this transition the sensitivity coefficient (Flambaum and Dzuba 2009) is $K_{E2,E3}^{\alpha} = -6.95$ which implies

$$ \frac{\nu_{Yb-E3}}{\nu_{Yb-E2}} = \alpha_{EM}^{-6.95}. \qquad (138) $$
* **Yb vs caesium-133.** The ¹S₀ - ³P₀ transition in neutral ¹⁷¹Yb serves as a basis for frequency standards (Brown et al. 2017). Its sensitivity to α<sub>EM</sub> was computed by Safronova et al. (2018c). McGrew et al. (2019a) compared the ¹S₀ - ³P₀ transition of ¹⁷¹Yb to an international collection of national primary and secondary frequency standards using satellite time and frequency transfer to get

$$ \frac{d}{dt} \ln \left( \frac{v_{Yb}}{v_{Cs}} \right) = (-4.9 \pm 3.6) \times 10^{-17} \text{ yr}^{-1} \quad (139) $$

and Table 6 gives that it behaves

$$ \frac{v_{Rb}}{v_{Cs}} \propto (g_{Cs}\bar{\mu})^{-1} \alpha_{EM}^{-2.52}. $$

## Atomic dyprosium

It was suggested in Dzuba et al. (1999a, 2003) (see also Dzuba and Flambaum (2010) for a computation of the transition amplitudes of the low states of dyprosium) that the electric dipole (E1) transition between two nearly degenerate opposite-parity states in atomic dyprosium should be highly sensitive to the variation of α<sub>EM</sub>. It was then demonstrated (Nguyen et al. 2004) that a constraint of the order of 10<sup>-18</sup>/yr can be reached. The frequencies of nearly of two isotopes of dyprosium were monitored over a 8 months period (Cingöz et al. 2008) showing that the frequency variation of the 3.1-MHz transition in ¹⁶³Dy and the 235-MHz transition in ¹⁶²Dy are 9.0 ± 6.7 Hz/yr and -0.6 ± 6.5 Hz/yr, respectively. As seen from Table 6, the Dy/Cs frequency comparison is over six orders of magnitude more sensitive to variation of α<sub>EM</sub> than to variation of (μ, X<sub>q</sub>) so that it provides a constraints on α<sub>EM</sub> alone

$$ \frac{\dot{\alpha}_{EM}}{\alpha_{EM}} = (-2.7 \pm 2.6) \times 10^{-15} \text{ yr}^{-1}, \quad (140) $$

at 1σ level, without any assumptions on the constancy of other constants. Leefer et al. (2013) used the 6735.5-MHz transition in ¹⁶⁴Dy instead of the 3.1-MHz transition in ¹⁶³Dy to reach

$$ \frac{\dot{\alpha}_{EM}}{\alpha_{EM}} = (-5.8 \pm 6.9) \times 10^{-17} \text{ yr}^{-1}. \quad (141) $$

## Summary

The experimental constraints on laboratory clock comparison are summarized in Table 9 in which we have included their dependencies in the QED primary parameters using the data from Table 6. In the end all frequency comparisons depend only on the 4 parameters (α<sub>EM</sub>, μ̄, g<sub>Cs</sub>, g<sub>Rb</sub>). Indeed, they are not independent but the set of data can be used to derived constraints on their independent time variation.
### 5.1.3 Constraints on primary parameters

#### Constraint on $\alpha_{EM}$ alone

Since four clock comparisons listed in Table 9 depend on $\alpha_{EM}$ only, it can be constrained independently of the 3 others primary parameters.

<figure><img src="image_20.png" /><figcaption>**Fig. 13** Synthetic constraints on ($\alpha_{EM}$, $g_{Cs}\bar{\mu}$) from the atomic clock constraints summarized in Table 9. The upper left-plot reproduces Huntemann et al. (2014), note however that it depicts $g_{Cs}\bar{\mu}$ instead of $\mu = m_p/m_e$. The contour plots indicate $1\sigma$ and $2\sigma$ constraints and the black point the best-fit. The data used combine comparisons of optical clocks [red: Hg-Al (Rosenband et al. 2008); Purple: Dy 162–164 (Leefer et al. 2013)] and comparison of optical clocks to caesium [Green: Hg-Cs (Fortier et al. 2007); Blue: Yb(E2)-Cs (Tamm et al. 2014); Orange: Yb(E3)-Cs (Huntemann et al. 2014)]. The upper-right plot updates the Yb(E3)-Cs data by Lange et al. (2021) and adds the Yb-Cs data [purple] by McGrew et al. (2019a). The lower-left plot compares the Yb(E3)-Cs data by Huntemann et al. (2014) and Lange et al. (2021) and adds the Sr-Cs data [Brown] by Schwarz et al. (2020). The lower-right plot zooms on the dashed region and adds the constraints (142) on $\alpha_{EM}$. The black dot and contour lines correspond to the best-fit, $1\sigma$ and $2\sigma$ constraints obtained from all data except Sr-Cs and the $\alpha_{EM}$ constraint while the red dot is the best fit for all data but the $\alpha_{EM}$ constraint. The red ellipses correspond to $1\sigma$ and $2\sigma$ constraints obtained from all data</figcaption></figure>
The aluminium-mercury clocks comparison (Rosenband et al. 2008) depends only on $\alpha_{EM}$ so that the constraint (131) translates to $\alpha_{EM} = (-1.6 \pm 2.3) \times 10^{-17} \text{yr}^{-1}$ on $\alpha_{EM}$ alone without any use of complementary data. Since its sensitivity to $\alpha_{EM}$ is 6 order of magnitude higher than other sensitivities, see Eq. (141), the dyprosim comparison (Leefer et al. 2013) gives in practice $\alpha_{EM} = (-5.8 \pm 6.9) \times 10^{-17} \text{yr}^{-1}$. Then, the Yb:E2-E3 comparison by Lange et al. (2021) implies the bound $\alpha_{EM} = (1 \pm 1.1) \times 10^{-18} \text{yr}^{-1}$. The improvements by Filzinger et al. (2023) translate to

$$ \frac{\alpha_{EM}}{\alpha_{EM}} = (1.8 \pm 2.5) \times 10^{-19} \text{yr}^{-1}. \qquad (142) $$

Combining the independent measurements does not improve this latter constraint, which can be considered as the sharpest constraint on the independent variation of $\alpha_{EM}$ today.

### Constraints on other constants

Several analysis have been performed to extract constraints on $\bar{\mu}$ and $X_q$. Their results have evolved with the improvement of experimental data that have been combined in different ways. We refer to Fig. 13 for a up-to-date status on the constraints on $(\alpha_{EM}, g_{Cs}\bar{\mu})$.

From their result on Rb/Cs, Guena et al. (2012) derived $d \ln[\alpha_{EM}^{-0.49} X_q^{-0.021}] = (-1.36 \pm 0.91) \times 10^{-16} \text{yr}^{-1}$. Combined with the constraints on $\alpha_{EM}$ by Rosenband et al. (2008), they concluded

$$ K_{Cs}^q \dot{X}_q / X_q = (1.4 \pm 0.9) \times 10^{-17} \text{yr}^{-1}. \qquad (143) $$

This constraint was then used by Huntemann et al. (2014) to analyse their Yb(E3)-Cs results fitted together with the constraints by Rosenband et al. (2008) and Leefer et al. (2013) to conclude that $\alpha_{EM}/\alpha_{EM} = (-0.20 \pm 0.20) \times 10^{-16} \text{yr}^{-1}$ and $\dot{\mu}/\bar{\mu} = (-0.5 \pm 1.6) \times 10^{-16} \text{yr}^{-1}$. Godun et al. (2014) follows the same route but assumes the sensitivity to $X_q$, was zero to get the similar results, $\alpha_{EM}/\alpha_{EM} = (-0.7 \pm 2.1) \times 10^{-17} \text{yr}^{-1}$ and $\dot{\mu}/\bar{\mu} = (0.2 \pm 1.1) \times 10^{-16} \text{yr}^{-1}$. This series of analysis were summarized in Fig. 3 of Huntemann et al. (2014). Combining with the constraint on $\alpha_{EM}$ by Rosenband et al. (2008) and on $X_q$ by Ashby et al. (2018), McGrew et al. (2019b) concluded that $\dot{\mu}/\bar{\mu} = (5.3 \pm 6.5) \times 10^{-17} \text{yr}^{-1}$. To finish, Schwarz et al. (2020) combined (143) with the constraint on $\alpha_{EM}$ by Godun et al. (2014) and Huntemann et al. (2014), to conclude that $\dot{\mu}/\bar{\mu} = (-6.9 \pm 6.5) \times 10^{-17} \text{yr}^{-1}$.

### Clock constraints on time variation of $\alpha_{EM}$, $\bar{\mu}$ and $X_q$

Figure 13 reproduces and updates Fig. 3 of Huntemann et al. (2014) including the new constraints by Lange et al. (2021), McGrew et al. (2019a), Schwarz et al. (2020) and Filzinger et al. (2023). An efficient way to disentangle the constraints is through 3 steps.

1. First, the comparisons of optical clocks set the constraint (142) on $\alpha_{EM}$ alone.
2. Then, the Rb/Cs experiments constrain $g_{Rb}/g_{Cs}$ to get

$$ \begin{aligned} \frac{d}{dt} \ln \left( \frac{g_{Rb}}{g_{Cs}} \right) &= \frac{d}{dt} \ln \left( X_q^{-0.021} \right) \qquad (144) \\ &= (-1.16 \pm 0.61) \times 10^{-16} \text{ yr}^{-1}, \end{aligned} $$

using Abgrall et al. (2015).

3. To finish, the comparisons of optical clocks to Cs-133 provide constraints on $g_{Cs}\bar{\mu}$. Given the bound (142), the combined analysis of the data by Fortier et al. (2007), Tamm et al. (2014), Lange et al. (2021), McGrew et al. (2019a), Schwarz et al. (2020) gives

$$ \begin{aligned} \frac{d}{dt} \ln(g_{Cs}\bar{\mu}) &= \frac{d}{dt} \ln(X_q^{0.002}\bar{\mu}) = \frac{d}{dt} \ln(X_q^{-0.046}X_e) \qquad (145) \\ &= (0.42 \pm 0.13) \times 10^{-16} \text{ yr}^{-1}. \end{aligned} $$

4. These three latter bounds can be combined to get a constraints on $\dot{X}_q$ and $\dot{\bar{\mu}}$ or $\dot{X}_e$.

As a conclusion, the global fit of the data discussed above allows us to set the following constraints

$$ \left. \frac{\dot{\alpha}_{EM}}{\alpha_{EM}} \right|_0 = (1.8 \pm 2.5) \times 10^{-19} \text{ yr}^{-1}, \qquad (146) $$

$$ \left. \frac{\dot{\bar{\mu}}}{\bar{\mu}} \right|_0 = (3.09 \pm 1.42) \times 10^{-17} \text{ yr}^{-1}, \qquad (147) $$

$$ \left. \frac{\dot{X}_q}{X_q} \right|_0 = (5.5 \pm 2.9) \times 10^{-15} \text{ yr}^{-1}, \qquad (148) $$

$$ K_{Cs}^q \left. \frac{\dot{X}_q}{X_q} \right|_0 = (1.10 \pm 0.58) \times 10^{-17} \text{ yr}^{-1}, \qquad (149) $$

$$ \left. \frac{\dot{X}_e}{X_e} \right|_0 = (2.96 \pm 1.34) \times 10^{-16} \text{ yr}^{-1}, \qquad (150) $$

where we recall that $\bar{\mu} = m_e/m_p$, $X_{e,q} = m_{e,q}/\Lambda_{QCD}$ and the sensitivity coefficient $K_{Cs}^q = 0.002$, see Eq. (108).

### 5.1.4 Fiber-linked networks and clocks in space

To extend the analysis of the previous section, on can compare clocks at different locations since they can be networked online using either satellites (Schiller et al. 2006; Magnani et al. 2019)—see Belenchia et al. (2022), Alonso et al. (2022) for a review and a roadmap on quantum physics in space—, fibre links (Roberts et al.
2020), or offline via time-stamping of measurements (Wcisło et al. 2018)— see Savalle et al. (2019b) for a discussion of spacetime separated clock experiments to test variations of fundamental constants. It has recently been implemented by Filzinger et al. (2025) to search for ultralight dark matter.

### Fiber-linked comparisons

Roberts et al. (2020) searched for transient variations of $\alpha_{EM}$ using data from the European network of fiber-linked optical clocks based on Sr, Hg and Yb$^+$ atoms located in France, Germany and United Kingdom. Thanks to the use of fiber links to perform the comparisons, the stability of the measurements is limited by the one of the clocks themselves (Lisdat et al. 2016). It was concluded that $\Delta\alpha_{EM}/\alpha_{EM} < 5 \times 10^{-17}$ on a duration of $10^3$ s. The QSNET project (Barontini et al. 2021, 2022) is a network of clocks that will include existing Sr, Yb$^+$ and Cs atomic clocks at the National Physical Laboratory (NPL) in London and several new clocks ($N_2^+$ molecular ion clock, CaF molecular optical lattice clock and a Cf highly charged ion clock).

### Atomic clocks in space (ACES)

An improvement of at least an order of magnitude on current constraints can be achieved in space. The PHARAO/ACES project (Salomon et al. 2001; Reynaud et al. 2009) (Projet d'Horloge Atomique par Refroidissement d'Atomes en Orbite) of the European Spatial Agency combines laser cooling techniques and a microgravity environment in orbit. It consists of a Cs clock installed on the outside pallet of the Columbus module of the ISS that can be compared to ground clocks using two-way microwave links. It aims at performing redshift test at the level of $2 - 3 \times 10^{-6}$ during 1.5 to 3 years. Its performances are evaluated in Savalle et al. (2019a).

The Galileo gravitational redshift test (Delva et al. 2018, 2019), GREAT, has been comparing H-maser clocks onboard of eccentric satellites to ground clocks to conclude that $\beta_{H-maser} = (0.19 \pm 2.46) \times 10^{-5}$ (Delva et al. 2019); see Eq. (379) below for the definition of the redshift parameter. The proposal FOCOS (Derevianko et al. 2022) aims to improve this constraint to $10^{-9}$.

The SAGAS (Search for anomalous gravitation using atomic sensor) project (Reynaud et al. 2009; Wolf et al. 2009) aims at flying highly sensitive optical atomic clocks and cold atom accelerometers on a Solar system trajectory on a time scale of 10 years. It could test the constancy of the fine-structure constant along the satellite worldline, which, in particular, can set a constraint on its spatial variation of the order of $10^{-9}$.

Among these recent developments we shall also mention the proposal by Terno et al. (2023) to suppress the first-order Doppler effect which dominates the weak gravitational signal and the mission (Schkolnik et al. 2023) of optical atomic clocks aboard the space station (OACESS).

### 5.1.5 Experimental constraints from molecular transitions

As we have seen, only atomic clocks based on a hyperfine transition involve $\bar{\mu}$ and they all share the same sensitivity which makes it difficult to extract a model-
independent constraints on the variation of $\bar{\mu}$. Molecules and precision molecular spectroscopy provide pathway toward better constraint on the variation of $\bar{\mu}$ that are complementary to the constraints from atomic clocks; see e.g., (Jansen et al. 2014) for a dedicated review.

## Generalities

The first to be considered were diatomic molecules since, as first pointed out by Thompson (1975), molecular lines can provide a test of the variation of $\mu$. The energy difference between two adjacent rotational levels in a diatomic molecule is inversely proportional to $Mr^{-2}$, $r$ being the bond length and $M$ the reduced mass, and the vibrational transition of the same molecule has, in first approximation, a $\sqrt{M}$ dependence. For molecular hydrogen $M = m_p/2$ so that the comparison of an observed vibro-rotational spectrum with a laboratory spectrum gives an information on the variation of $m_p$ and $m_n$. Comparing pure rotational transitions with electronic transitions gives a measurement of $\mu$. It follows that the frequency of vibro-rotation transitions is, in the Born-Oppenheimer approximation, of the form

$$v \approx E_I(c_{\text{elec}} + c_{\text{vib}}\sqrt{\bar{\mu}} + c_{\text{rot}}\bar{\mu}) \quad (151)$$

where $c_{\text{elec}}$, $c_{\text{vib}}$ and $c_{\text{rot}}$ are some numerical coefficients. Molecules such as $H_2$, HD or $NH_3$ have been extensively used in astrophysics (see Sect. 5.7 below).

## SF$_6$ transition

The comparison of the vibro-rotational transition in the molecule SF$_6$, using the transition is P(4)E$_0$ in the $2v_3$ band, was compared to a caesium clock over a two-year period, leading to the constraint (Shelnikov et al. 2008)

$$\frac{d}{dt} \ln \left( \frac{v_{\text{SF6}}}{v_{\text{Cs}}} \right) = (1.9 \pm 0.12 \pm 2.7) \times 10^{-14} \text{yr}^{-1}, \quad (152)$$

where the second error takes into account uncontrolled systematics. Now, Table 6 gives

$$\frac{v_{\text{SF6}}}{v_{\text{Cs}}} \propto \bar{\mu}^{1/2} \alpha_{\text{EM}}^{-2.83} (g_{\text{Cs}}\bar{\mu})^{-1}.$$

It can be combined with the constraint (129) which enjoys the same dependence to caesium and (142) to infer that

$$\frac{\dot{\bar{\mu}}}{\bar{\mu}} = (-3.8 \pm 5.6) \times 10^{-14} \text{yr}^{-1}. \quad (153)$$

Combined with Eq. (145), we can obtain independent constraints on the time variation of $g_{\text{Cs}}$, $g_{\text{Rb}}$ and $\bar{\mu}$.

While this is indeed less stringent that the constraints from atomic clocks discussed in the previous section, molecular clocks are a fast developing field; see e.g. Oswald et al. (2022). The success of techniques such as laser cooling, trapping or coherent manipulations in atomic physics have motivated their extension to
molecules and the control of their quantum state offers new direction too test fundamental physics and in particular fundamental constants (Mitra et al. 2022). This has been extended to the proposal to consider optical clocks based on molecular vibrations (Hanneke et al. 2021).

## KRb molecule

To date, the best constraint has been obtained from KRb (Kobayashi et al. 2019). Using a transition between nearly a degenerate pair of vibrational levels each associated with a different electronic potential of ultracold diatomic alkali KRb molecules that enjoys a large sensitivity coefficient, $\Delta v/v \simeq (14\,890 \pm 60)\Delta \bar{\mu}/\bar{\mu}$, it was concluded that

$$ \frac{\dot{\bar{\mu}}}{\bar{\mu}} = (0.3 \pm 1.0) \times 10^{-14} \text{ yr}^{-1}. \qquad (154) $$

## Other proposals

There is a large activity to increase the stability and sensitivity of molecular clocks, both theoretically and experimentally. We list s series of proposals and latest developments on this rich and active field of research.

* **Simple molecules**, such as $H_2^+$ and HD$^+$, offer the possibility of accurate theoretical computations while many experiments exist with high precision spectroscopy (Ubachs et al. 2016b; Biesheuvel et al. 2016; Alighanbari et al. 2020) making them good candidates for test of $\mu$ variation (Schiller and Korobov 2005; Schiller et al. 2014). They are also of importance for astrophysics; see Sect. 5.7 below.

* **Diatomic molecules.** Flambaum and Kozlov (2007b) estimated the enhanced sensitivity of very close narrow levels of different nature that exist in diatomic molecules due to cancelation between the fine structure and vibrational intervals in the electronic ground state, such as $Cl_2^+$, CuS, IrC, SiBr, HfF$^+$, LaS, LuO, HfF$^+$ and $^7$LiH-$^6$LiH (Constantin 2015) for which the vibrorotational transition in the $v = 0 \rightarrow 1$ band has a sensitivity of $-585$. We refer to Ubachs et al. (2016a) for a review on $H_2$, Kajita (2008) for $^2$4MgH and $^{40}$CaH, Kajita and Moriwaki (2009) for $CaH^+$ or $^{207}$Pb$^{19}$F that enjoys a pair of closely spaced levels of opposite levels (Flambaum et al. 2013). We shall also mention transitions with hyperfine-structures, for which the sensitivity to $\alpha_{EM}$ can reach 600 for instance $^{139}$La$^{32}$S or silicon monobrid (Beloy et al. 2010) that allows one to constrain $\alpha_{EM}\bar{\mu}^{-1/4}$.

* **Cold diatomic molecules**: 2-photons Raman transitions in lattice-confined photo-associated $Sr_2$ molecules (Zelevinsky et al. 2008; Reinaudi et al. 2012) or $Cs_2$ molecules (DeMille et al. 2008; Beloy et al. 2011a) have been considered.

* **Non-polar molecular ions of homonuclear diatomics**, such as $N_2^+$ or $O_2^+$. Kajita (2017a, 2017b) estimates that the Stark and Zeeman shifts in the transition frequencies of the $O_2^+$ molecular ion can allow experiments to reach $\Delta\mu/\mu < O(10^{-17}$ since the sensitivities $\Delta\ln v/\delta\ln\mu$ for the transitions $X^2\Pi_{1/2} v = 0 \rightarrow v'$ ($v' = 1, 4, 8$) are of the order $-0.48$ at frequencies (56.5, 219, 421.9) THz. It can even reach 140 for the transition $X^2\Pi_{1/2}v = 21 \rightarrow$
$a^4\Pi_{1/2}\nu = 0$ at 2.7 THz (Hanneke et al. 2016) leading Carollo et al. (2018) to estimate one could reach $\Delta\mu/\mu \sim O(10^{-18})$.

* *Molecular ions.* Beloy et al. (2011b) have computed the sensitivities of the NH$^+$ rotational spectrum to variations of ($\alpha_{EM}, \mu$).

* *Polar molecules* with deep potential such as TeH$^+$ allow for high sensitivity dipole-allowed vibrational transitions. Stollenwerk et al. (2018) estimated that with enough vibrational cooling laser power to saturate all the transitions after one can reach $\Delta\mu/\mu \sim 3.6 \times 10^{-17}$, as confirmed by Kokish et al. (2018). They mention that simulations also support the potential for fluorescence state read-out of TeH$^+$ suggesting the possibility of searching for $\mu$-variation using multi-ion spectroscopy on laser-coolable polar species.

* *Dihalogens and hydrogen halides:* Pašteka et al. (2015) identified strong candidates among HBr$^+$, HI$^+$, Br$_2^+$, I$_2^+$, IBr$^+$, ICl$^+$, and IF$^+$.

* *Tunneling-Rotational Transitions* have enhanced sensitivities. Ethylene glycol (C$_2$H$_6$O$_2$) in its ground conformation has such a transition with the frequency about 7 GHz. Since tunneling and rotational energies have different dependencies on $\bar{\mu}$, the spectrum is highly sensitive to the possible its variation with sensitivities ranging from -17 to +18 (Viatkina and Kozlov 2014). Ethylene glycol has been detected in the interstellar medium.

* *Internal rotor molecules* in molecules exhibiting hindered internal rotation, such as methyl mercaptan (CH$_3$), have torsion-rotation transitions with enhanced sensitivities to a variation of $\bar{\mu}$(Jansen et al. 2013). This enhancement occurs due to a cancellation of energies associated with the torsional and rotational degrees of freedom of the molecule. It is also exceptionally large in methanol that is detected in quasar absorption spectra.

* *Polyatomic molecules.* Owens et al. (2015) computed the sensitivities of $^{14}$NH$_3$, $^{15}$NH$_3$, $^{14}$ND$_3$, and $^{15}$ND$_3$ allowing Owens et al. (2016) to show that the sensitivity of amonia vibro-rotation lines can reach -16.738; see also Jansen et al. (2014). Santamaria et al. (2014) proposed an experiment to constrain, over a-few-year timescale, the fractional temporal variation of $\mu$ at the level of $10^{-15}$/yr by means of a spectroscopic frequency measurement on a beam of cold CF$_3$H molecules. Kozlov and Levshakov (2011) stressed that the tunneling transition of H$_3$O$^+$ occurs in the far-infrared and that such transitions are observed within the interstellar medium in the Milky Way as well as nearby galaxies. Hence, they computed its sensitivity to a variation of $\mu$ and argued it can be used as an independent target to test hypothetical changes in $\mu$ in different ambient conditions of high (terrestrial) and low (interstellar medium) matter densities. They also considered H$_2$O$_2$ (Kozlov 2011) to have exhibited a transition at 14.8 GHz with a sensitivity of 37 that can be observed astrophysically (Bergman et al. 2011). Kozlov (2013) also mentioned that the rovibrational spectrum of polyatomic molecules with $\Pi$-electronic ground state can be strongly modified by the Renner-Teller effect, leading increased sensitivities.

* Precision measurement of the scattering lengths in Bose-Einstein condensate and Feshbach molecular experiments were also discussed (Chin and Flambaum
2006) and argued to reach a $10^{-15} - 10^{-16}$ level on the variation of $\mu$ (Gacesa and Côté 2014).

While there is still a gap of 2 to 3 orders of magnitude on the constraint on $\dot{\mu}/\mu$ between molecular and atomic clocks, the former is expected to become competitive in a near future. Molecular structure and dynamics offer rich energy scales that are at the heart of new protocols in precision measurement and quantum information science (Kondov et al. 2019) and many candidates with high sensitivity have been singled out. These clocks will allow one to test both a drift and rapid oscillations and will complete the constraints obtained from atomic clocks. This will complement the use of molecular spectra in astrophysics by providing both theoretical computations of the sensitivity coefficients and high-precision laboratory spectra for comparison.

### 5.1.6 Nuclear clocks as a possible new system

The transition frequency of nuclear energy levels are generally outside the laser accessible range by several orders of magnitude. It has however been noticed that the nuclear transitions of an optical clock based on a very narrow ultraviolet nuclear transition between the low lying $^{2}$29mTh isomer (i.e., a long-lived excited nuclear state ($10^3 - 10^4$ s) to its ground state with predicted energy of $7.8 \pm 0.05$ eV and $\Delta E/E \sim 10^{-20}$ (Beck et al. 2007; Beeks et al. 2021) is an exception. The existence of the isomere was confirmed by von der Wense et al. (2016). The transition has been discovered experimentally (Tiedau et al. 2024; Elwell et al. 2024) in CaF$_2$ thorium-doped crystals at an energy of 8.4 eV, i.e., at about $10\sigma$ of the theoretical predictions. State-resolved laser spectroscopy at the $10^{-12}$ precision level was recently reported by Zhang et al. (2024) and the first constraints on the ultra-light dark matter models—see Sect. 2.3.4 for description—have been obtained by Fuchs et al. (2025).

On the one hand, it has attracted interest in the design of a clock based on such a nuclear transition (Peik and Tamm 2003; Campbell et al. 2012) who showed that such a clock can reach $\delta\omega/\omega \sim 10^{-19}$. On the other hand, using a Walecka model for the nuclear potential, Flambaum (2006) concluded that the sensitivity of the transition to the fine-structure constant and quark mass was typically

$$ \frac{\delta\omega}{\omega} \sim 10^5 \left( 4\frac{\delta\alpha_{EM}}{\alpha_{EM}} + \frac{\delta X_q}{X_q} - 10\frac{\delta X_s}{X_s} \right). $$

This roughly provides a five orders of magnitude amplification, which can lead to a constraint at the level of $10^{-23}$/yr on the time variation of $X_q$. Such a method is promising and would offer different sensitivities to systematic effects compared to atomic clocks. However, this sensitivity is not clearly established since different nuclear calculations do not agree (Berengut et al. 2009) and Hayes and Friar (2007) questioned the result on the basis of the Feynman-Hellmann theorem—that relates the derivative of the total energy with respect to a parameter to the expectation value
of the derivative of the Hamiltonian with respect to that same parameter. One expects that

$$ \frac{\Delta\omega}{\omega} = K\frac{\Delta\alpha_{\text{EM}}}{\alpha_{\text{EM}}} \qquad K = \frac{\Delta V_c}{\omega}. $$

$\Delta V_c$, the change in the Coulomb energy, is the central physical quantity to understand the high sensitivity since it derives from an almost perfect cancelation of its change, $\Delta V_c = E_c^m - E_c^{\text{ground}} \sim -1$ MeV. Hayes and Friar (2007) argued that no sensitivity to $\alpha_{\text{EM}}$ should arise while He and Ren (2008) confirmed that it shall be $-9.2 \times 10^4$ pointing out the importance of the correlation between the nuclear and electromagnetic interactions. The small difference between the Coulomb energies of the two states, which both are of the order $10^9$ eV, is the cause of the large amplification of the sensitivity to a $\alpha_{\text{EM}}$ variation. Litvinova et al. (2009), Feldmeier et al. (2017) used Hartree-Fock and Hartree-Fock-Bogoliubov calculations to compute the Coulomb and kinetic energies between the $3/2^+$ and $5/2^+$ states. It concluded to the limited precision of the nuclear model calculations. Flambaum et al. (2009a) then estimated the contribution of the polarization of the Coulomb energy to the spacing between the two states as a function of the nuclear deformation again concluding to a sensitivity of order $10^4$. Berengut et al. (2009) proposed a method to extract the $\alpha_{\text{EM}}$ dependence and it was concluded by Thielking et al. (2018), based on Litvinova et al. (2009), that

$$ \frac{\Delta E_c}{1 \text{ MeV}} = -485 \left[ \frac{\langle r_{229\text{m}}^2 \rangle}{\langle r_{229}^2 \rangle} - 1 \right] + 11.6 \left[ \frac{Q_0^m}{Q_0} - 1 \right] \approx -0.29 \pm 0.0043, $$

with $\langle r_{229}^2 \rangle$ the nuclear mean-square charge radius and $Q_0$ the electric quadrupole moment. This value is dominated by the experimental uncertainty of $Q_0^m/Q_0$ of about 4%. It follows that the nuclear properties of both states have to be determined precisely to derive the actual sensitivity (see e.g., Safronova et al. (2018b) for $\langle r_{229}^2 \rangle$). We refer to Thirolf et al. (2019a, 2019b), Peik et al. (2021) for a detailed description of these modelizations and for the prospective of nuclear clocks for fundamental physics. We refer to Caputo et al. (2024) for further developments. Note that, assuming a prolate spheroid nucleus, Beeks et al. (2024) estimated the sensitivity of the nuclear transition frequency to variations of $\alpha_{\text{EM}}$ to be $K_x = 5900 \pm 2300$, i.e., a three orders of magnitude enhancement over atomic clock schemes based on electron shell transitions.

To finish let us mention that Fadeev et al. (2022) studied the sensitivity of Mössbauer transitions on $X_q$ to conclude that

$$ \frac{\delta\omega}{\omega} = 1.45 \left( \frac{\delta\alpha_{\text{EM}}}{\alpha_{\text{EM}}} - 1 \right) \frac{\delta m_q}{m_q} $$

pointing to a sensitivity of $10^4$ for $^{229}\text{Th}$ and $10^3$ for $^{235}\text{U}$.
### 5.1.7 Future evolutions

The constraints from clock comparisons of is mostly determined by the uncertainties of the clocks, their sensitivities to the variation of the constants, and the time interval over which the comparisons are made. All these factors have been continuously improved in the past years. Hence one has 3 strategies:

1. repeat experiments on a longer time scale.

2. improve existing clocks. The accuracy of atomic clocks has improved by a factor $10^3$ in the past 15 years to reach a fractional frequency uncertainty of two parts in $10^{18}$. The most rapid improvement is expected to come from optical to optical clock comparison—see e.g., Aeppli et al. (2024) who reported an optical lattice clock with a total systematic uncertainty of $8.1 \times 10^{-19}$—while optical to microwave comparison being limited by the ultimate accuracy of microwave clocks. Comparison of different clocks beyond $10^{-18}$ will become more challenging due to the sensitivity to the environment, including temperature and gravitational potential; for a review see e.g., Ludlow et al. (2015).

3. develop new clocks with higher sensitivities. First, a good candidate are highly charged ions (Flambaum and Porsev 2009; Berengut et al. 2010a; Kozlov et al. 2018; Rehbehn et al. 2021) and in particular optical of heavy actinides from uranium to einsteinium (Allehabi et al. 2024; Dzuba and Flambaum 2024). Many systems, offering transitions are between ground and excited metastable states of the ions, are promising candidates for optical clocks. These clocks are less sensitive to external perturbations and their sensitivity to variation of $\alpha_{EM}$ is enhanced due to larger relativistic effects. Molecular transitions, described in Sect. 5.1.5 have significantly improved and many candidates are being studied. Then we have discussed in Sect. 5.1.6 the developments of nuclear clocks.

We refer to the reviews (Safronova et al. 2018a; Safronova 2019) on these three issues. Indeed this goes together with the developments of quantum sensors (Tsai et al. 2023) and algorithm (Zaheer et al. 2025). Besides, there is a strong developments to compare clocks at different spacetime positions, as discussed in Sect. 5.1.4, either thanks to clock network or in space; see e.g., Derevianko et al. (2022), Barontini et al. (2022). This allows one to probe spacetime correlations and is the only possibility of detecting transient events linked to macroscopic dark objects. These developments are motivated by the search of light matter candidate and the tests of General Relativity. As explained on Fig. 5 various modes of operation target different physical sectors.

## 5.2 Macroscopic bodies and chemistry

Let us briefly mention the idea that a variation of constants would induce the size and shape of material objects.

Stadnik and Flambaum (2016a) consider the effect of a time variation of the constants on laser interferometer since it would alter the accumulated phase, $\phi =$
$\omega L/c$ of the light beam inside. Indeed both the transition frequency $\omega$ and the size of the interferometer $L \propto a_B \propto \hbar^2/m_e 2e^2$ would vary so that $\delta\varphi \simeq \varphi\delta\alpha_{EM}/\alpha_{EM}$ for an atomic transition.

Similarly, Pašteka et al. (2019) remarked that in non-relativistic physics the size of molecules and solids is proportional to the Bohr radius. Since this dependence cancels out in the ratio of the sizes, the individual dependence of different compounds on $\alpha_{EM}$ arises from the difference in the relativistic effects. It led them to investigate the variation of crystal lattice parameters and molecular bond lengths due to variation of the fine structure constant and the proton-to-electron mass ratio for selected solid state and molecular systems.

Ellis et al. (2020) discussed the change in macroscopic objects and the influence on the structure of molecules, including bond lengths and geometry. In the case of the water molecules their dependencies on ($\alpha_{EM}, \bar{\mu}$) was determined from heavy numerical simulations (King et al. 2010). This was extrapolated to the structure of organic molecules such as DNA. Following the statement by Calladine and Drew (1992) that “the distant between adjacent sugars or phosphates in the DNA chain must be between 5.5 and 6.5 Angstroms” for the DNA helix structure to exist. They claim one would need $\delta(\ell_{A+T}/\ell_{G+C})$ shall be smaller than 10%, where $\ell_{A+T}$ and $\ell_{G+C}$ stand for the lengths of the A-T and G-C bonds.

Trachenko (2023) investigated the effects of a variation of the constants on condensed matter and liquid physics, focusing on viscosity and diffusion in order to translate the range of bio-friendly viscosity and diffusion on the range of fundamental constants which favor cellular life.

Indeed those systems are not competitive with quantum systems but they may be of interest for discussions circling around the anthropic principle; see Sect. 9.

## 5.3 The Oklo phenomenon

### 5.3.1 A natural nuclear reactor

Oklo is the name of a town in the Gabon republic (West Africa) where an open-pit uranium mine is situated. About $1.8 \times 10^9$ yr ago (corresponding to a redshift of ~0.14 with the cosmological concordance model), in one of the rich vein of uranium ore, a natural nuclear reactor went critical, consumed a portion of its fuel and then shut a few millions years later (see, e.g., Uzan and Leclercq 2008; Davis et al. 2014 for more details). This phenomenon was discovered by the French Commissariat à l'Énergie Atomique (CEA) in 1972 while monitoring for uranium ores (Naudet 2000). Sixteen natural uranium reactors have been identified. Among the 15-17 reaction zones, well studied reactors include the zone RZ2 (about 60 boreholes, 1800 kg of $^{235}$U fissioned during $8.5 \times 10^5$ yr) and zone RZ10 (about 13 bore-holes, 650 kg of $^{235}$U fissioned during $1.6 \times 10^5$ yr). As reviewed by Davis et al. 2014, the age of the phenomena is constrained by the facts that (1) the stabilisation of the Oklo geological basement happened not earlier 2.7 Gyr ago, (2) the geological age of the local Francevillian sediments are estimated to $2.265 \pm 0.15$ Gyr and (3) the Great Oxydation, that happened 2.2 Gyr ago, during
which cyanobacteria triggered the increase of atmospheric oxygen by a factor ~100 which allowed for uranium to be converted to a soluble form and hence transported and precipitated.

The existence of such a natural reactor was predicted by Kuroda (1956) who showed that under favorable conditions, a spontaneous chain reaction could take place in rich uranium deposits. Indeed, two billion years ago, uranium was naturally enriched (due to the difference of decay rate between $^{235}$U and $^{238}$U) and $^{235}$U represented about 3.68% of the total uranium (compared with 0.72% today and to the 3–5% enrichment used in most commercial reactors). Besides, in Oklo the conditions were favorable: (1) the concentration of neutron absorbers, which prevent the neutrons from being available for the chain fission, was low; (2) water played the role of moderator (the zones RZ2 and RZ10 operated at a depth of several thousand meters, so that the water pressure and temperature was close to the pressurized water reactors of 20 Mpa and 300° Cand slowed down fast neutrons so that they can interact with other $^{235}$U and (3) the reactor was large enough so that the neutrons did not escape faster than they were produced. Typically, the reaction zones are lens shaped layers about 10 m long, 10 m wide and up to 0.8 m thick. It is estimated that the Oklo reactor powered 10 to 50 kW. This explanation is backed up by the substantial depletion of $^{235}$U as well as a correlated peculiar distribution of some rare-earth isotopes. These rare-earth isotopes are abundantly produced during the fission of uranium and, in particular, the strong neutron absorbers like $^{149}_{62}$Sm, $^{151}_{63}$Eu, $^{155}_{64}$Gd and $^{155}_{64}$Gd are found in very small quantities in the reactor.

From the isotopic abundances of the yields, one can extract information about the nuclear reactions at the time the reactor was operational and reconstruct the reaction rates at that time. One of the key quantity measured is the ratio $^{149}_{62}$Sm/$^{147}_{62}$Sm of two light isotopes of samarium, which are not fission products. This ratio of order of 0.9 in normal samarium, is about 0.02 in Oklo ores. This low value is interpreted (Shlyakhter 1976) by the depletion of $^{149}_{62}$Sm by thermal neutrons produced by the fission process and to which it was exposed while the reactor was active. The capture cross-section of thermal neutron by $^{149}_{62}$Sm

$$n + _{62}^{149}Sm \rightarrow _{62}^{150}Sm + \gamma \qquad (155)$$

is dominated by a capture resonance of a neutron of energy of about 0.1 eV ($E_r$ = 97.3 meV today). The existence of this resonance is a consequence of an almost cancellation between the electromagnetic repulsive force and the strong interaction.

Shlyakhter (1976) pointed out that this phenomenon can be used to set a constraint on the time variation of fundamental constants. His argument can be summarized as follows.

* First, the cross-section $\sigma_{(n,\gamma)}$ strongly depends on the energy of a resonance at $E_r$ = 97.3 meV.

* Geochemical data allow one to determine the isotopic composition of various element, such as uranium, neodymium, gadolinium and samarium. Gadolinium
and neodium allow one to determine the fluence (integrated flux over time) of the neutron while both gadolinium and samarium are strong neutron absorbers.

* From these data, one deduces the value of the averaged value of the cross-section on the neutron flux, $\hat{\sigma}_{149}$. This value depends on hypothesis on the geometry of the reactor zone.

* The range of allowed values of $\hat{\sigma}_{149}$ was translated into a constraint on $E_r$. This step involves an assumption on the form and temperature of the neutron spectrum.

* $E_r$ was related to some fundamental constant, which involve a model of the nucleus.

In conclusion, we have different steps, which all involve assumptions:

* Isotopic compositions and geophysical parameters are measured in a given set of bore-hold in each zone. A choice has to be made on the sample to use, in order, e.g., to ensure that they are not contaminated.

* With hypothesis on the geometry of the reactor, on the spectrum and temperature of the neutron flux, one can deduce the effective value of the cross-sections of neutron absorbers (such as samarium and gadolinium). This requires one to solve a network of nuclear reactions describing the fission.

* One can then infer the value of the resonance energy $E_r$, which again depends on the assumptions on the neutron spectrum.

* $E_r$ needs to be related to fundamental constant, which involves a model of the nucleus and high energy physics hypothesis.

We shall now detail the assumptions used in the various analyses that have been performed since the pioneering work of Shlyakhter (1976).

### 5.3.2 Constraining the shift of the resonance energy

*Cross-sections.*

The cross-section of the neutron capture (155) strongly depends on the energy of a resonance at $E_r = 97.3$ meV and is well described by a Breit-Wigner formula

$$ \sigma_{(n,\gamma)}(E) = \frac{g_0 \pi \hbar^2}{2 m_n E} \frac{\Gamma_n \Gamma_\gamma}{(E - E_r)^2 + \Gamma^2/4} \quad (156) $$

where $g_0 \equiv (2I + 1)(2s + 1)^{-1}(2I + 1)^{-1}$ is a statistical factor, which depends on the spin of the incident neutron $s = 1/2$, of the target nucleus $I$, and of the compound nucleus $J$. For the reaction (155), we have $g_0 = 9/16$. The total width $\Gamma \equiv \Gamma_n + \Gamma_\gamma$ is the sum of the neutron partial width $\Gamma_n = 0.533$ meV (at $E_r = 97.3$ meV and it scales as $\sqrt{E}$ in the center of mass) and of the radiative partial width $\Gamma_\gamma = 60.5$ meV. $^{155}_{64}$Gd has a resonance at $E_r = 26.8$ meV with $\Gamma_n = 0.104$ meV, $\Gamma_\gamma = 108$ meV and $g = 5/8$ while $^{157}_{64}$Gd has a resonance at $E_r = 31.4$ meV with $\Gamma_n = 0.470$ meV, $\Gamma_\gamma = 106$ meV and $g = 5/8$.
As explained in Sect. 5.3.1, this cross-section cannot be measured from the Oklo data, which allow one to only measure its value averaged on the neutron flux $n(v, T)$, $T$ being the temperature of the moderator. It is conventionally defined as

$$ \hat{\sigma} = \frac{1}{nv_0} \int \sigma_{(n,\gamma)} n(v, T) v dv, \qquad (157) $$

where the velocity $v_0 = 2200 \text{ m} \cdot \text{s}^{-1}$ corresponds to an energy $E_0 = 25.3 \text{ meV}$ and $v = \sqrt{2E/m_n}$, instead of

$$ \bar{\sigma} = \frac{\int \sigma_{(n,\gamma)} n(v, T) v dv}{\int n(v, T) v dv}. $$

When the cross-section behaves as $\sigma = \sigma_0 v_0 / v$, which is the case for nuclei known as “1/v-absorbers”, $\hat{\sigma} = \sigma_0$ and does not depend on the temperature, whatever the distribution $n(v)$. In a similar way, the effective neutron flux,

$$ \hat{\phi} = v_0 \int n(v, T) dv, \qquad (158) $$

differs from the true flux

$$ \phi = \int n(v, T) v dv. $$

However, since $\bar{\sigma}\phi = \hat{\sigma}\hat{\phi}$, the reaction rates are not affected by these definitions.

*Extracting the effective cross-section from the data.*

To “measure” the value of $\hat{\sigma}$ from the data, we need to solve the nuclear reactions network that controls the isotopic composition during the fission.

The samples of the Oklo reactors were exposed (Naudet 2000) to an integrated effective fluence $\int \hat{\phi} dt$ of about $10^{21} \text{neutron} \cdot \text{cm}^{-2} = 1 \text{ kb}^{-1}$. Assuming a steady state mode of operation (as in most of the works) during $3 \times 10^5 \text{ yr}$, this leads to a neutron flux density of $10^8 \text{neutron} \cdot \text{cm}^{-2} \text{s}^{-1}$ (typically 5 orders of magnitude smaller than in present day reactors) but one needs to keep in mind that the instantaneous flux can be much higher in the the reactor operated by pulses. It implies that any process with a cross-section smaller than 1 kb can safely be neglected in the computation of the abundances. This includes neutron capture by $^{144}_{62}\text{Sm}$ and $^{148}_{62}\text{Sm}$, as well as by $^{155}_{64}\text{Gd}$ and $^{157}_{64}\text{Gd}$. On the other hand, the fission of $^{235}_{92}\text{U}$, the capture of neutron by $^{143}_{60}\text{Nd}$ and by $^{149}_{62}\text{Sm}$ with respective cross-sections $\sigma_5 \simeq 0.6 \text{ kb}$, $\sigma_{143} \sim 0.3 \text{ kb}$ and $\sigma_{149} \ge 70 \text{ kb}$ are the dominant processes. It follows that the equations of evolution for the number densities $N_{147}$, $N_{148}$, $N_{149}$ and $N_{235}$ of $^{147}_{62}\text{Sm}$, $^{148}_{62}\text{Sm}$, $^{149}_{62}\text{Sm}$ and $^{235}_{92}\text{U}$ take the form

$$ \frac{dN_{147}}{\hat{\phi} dt} = - \hat{\sigma}_{147} N_{147} + \hat{\sigma}_{f235} y_{147} N_{235} \qquad (159) $$
$$ \frac{dN_{148}}{\hat{\phi}dt} = \hat{\sigma}_{147}N_{147} \qquad (160) $$

$$ \frac{dN_{149}}{\hat{\phi}dt} = - \hat{\sigma}_{149}N_{149} + \hat{\sigma}_{f235}y_{149}N_{235} \qquad (161) $$

$$ \frac{dN_{235}}{\hat{\phi}dt} = - \sigma_5 N_{235}, \qquad (162) $$

where $y_i$ denotes the yield of the corresponding element in the fission of $^{235}_{92}$U and $\hat{\sigma}_5$ is the fission cross-section. This system can be integrated under the assumption that the cross-sections and the neutron flux are constant and the result compared with the natural abundances of the samarium to extract the value of $\hat{\sigma}_{149}$ at the time of the reaction. Here, the system has been closed by introducing a modified absorption cross-section (Damour and Dyson 1996) $\sigma_5^*$ to take into account both the fission, capture but also the formation from the $\alpha$-decay of $^{239}_{94}$Pu. One can instead extend the system by considering $^{239}_{94}$Pu, and $^{235}_{92}$U (see Gould et al. 2006). While most studies focus on the samarium, Fujii et al. (2000) also includes the gadolinium even though

<table><caption>**Table 10** Summary of the analysis of the Oklo data</caption><thead><tr><th>Ore</th><th>neutron spectrum</th><th>Temperature (°C)</th><th>σ̂<sub>149</sub> (kb)</th><th>ΔE<sub>r</sub> (meV)</th><th>Ref</th></tr></thead><tbody><tr><td>?</td><td>Max</td><td>20</td><td>55 ± 8</td><td>0 ± 20</td><td>Shlyakhter (1976)</td></tr><tr><td>RZ2 (15)</td><td>Max</td><td>180–700</td><td>75 ± 18</td><td>-1.5 ± 10.5</td><td>Damour and Dyson (1996)</td></tr><tr><td>RZ10</td><td>Max</td><td>200–400</td><td>91 ± 6</td><td>4 ± 16</td><td>Fujii et al. (2000)</td></tr><tr><td>RZ10</td><td>Max</td><td>200–400</td><td>91 ± 6</td><td>-97 ± 8</td><td>Fujii et al. (2000)</td></tr><tr><td>–</td><td>Max+ept</td><td>327</td><td>91 ± 6</td><td>-45<sup>+7</sup><sub>-15</sub></td><td>Lamoreaux and Togerson (2004)</td></tr><tr><td>RZ2</td><td>Max+ept</td><td>200–300</td><td>73.2 ± 9.4</td><td>-5.5 ± 67.5</td><td>Petrov et al. (2006)</td></tr><tr><td>RZ2</td><td>Max+ept</td><td>200–300</td><td>71.5 ± 10.0</td><td>–</td><td>Gould et al. (2006)</td></tr><tr><td>RZ10</td><td>Max+ept</td><td>200–300</td><td>85.0 ± 6.8</td><td>–</td><td>Gould et al. (2006)</td></tr><tr><td>RZ2+RZ10</td><td>Max+ept</td><td>200–300</td><td>85.0 ± 6.8</td><td>7.2 ± 18.8</td><td>Gould et al. (2006)</td></tr><tr><td>RZ2+RZ10</td><td>Max+ept</td><td>200–300</td><td>85.0 ± 6.8</td><td>90.75 ± 11.15</td><td>Gould et al. (2006)</td></tr><tr><td>RZ10</td><td>Max+ept</td><td>200–300</td><td>85.0 ± 6.8</td><td>[-11.6, 26.0]</td><td>Davis et al. (2014)</td></tr><tr><td>RZ10</td><td>Max+ept</td><td>200–300</td><td>85.0 ± 6.8</td><td>[-101.9, -79.6]</td><td>Davis et al. (2014)</td></tr></tbody></table>

The principal assumptions to infer the value of the resonance energy $E_r$ are the form of the neutron spectrum and its temperature. Concerning the neutron spectrum, “Max” stands for Maxwellian and “Max+ept” for Maxwellian+epithermal
it is not clear whether it can reliably be measured (Damour and Dyson 1996), still giving similar results.

By comparing the solution of this system with the measured isotopic composition, one can deduce the effective cross-section. At this step, the different analyses (Shlyakhter 1976; Petrov 1978; Damour and Dyson 1996; Fujii et al. 2000; Lamoreaux and Togerson 2004; Petrov et al. 2006; Gould et al. 2006) differ from the choice of the data. The measured values of $\hat{\sigma}_{149}$ can be found in these articles. They are given for a given zone (RZ2, RZ10 mainly) with a number that correspond to the number of the bore-hole and the depth (e.g., in Table 2 of Damour and Dyson (1996), SC39-1383 means that we are dealing with the bore-hole number 39 at a depth of 13.83 m). Recently, another approach (Petrov et al. 2006; Gould et al. 2006) was proposed in order to take into account of the geometry and details of the reactor. It relies on a full-scale Monte-Carlo simulation (Briesmeister 2000) dealing with realistic geometries (but also operating temperature, amounts of uraninite, gangue and water) and a computer model of the reactor zone RZ2 (Petrov et al. 2006) and both RZ2 and RZ10 (Gould et al. 2006). It allows one to take into account the spatial distribution of the neutron flux.

## Determination of $E_r$

To convert the constraint on the effective cross-section, one needs to specify the neutron spectrum. In the earlier studies (Shlyakhter 1976; Petrov 1978), a Maxwell distribution,

$$n_{\text{th}}(v, T) = \left(\frac{m_n}{2\pi T}\right)^{3/2} e^{-\frac{m_n^2}{2k_B T}},$$

was assumed for the neutron with a temperature of 20°C, which is probably too small. Then $v_0$ is the mean velocity at a temperature $T_0 = m_n v_0^2 / 2k_B = 20.4^\circ\text{C}$. Damour and Dyson (1996), Fujii et al. (2000) also assumed a Maxwell distribution but let the moderator temperature vary so that they deduce an effective cross-section $\hat{\sigma}(R_r, T)$. They respectively restricted the temperature range to $180^\circ\text{C} < T < 700^\circ\text{C}$ and $200^\circ\text{C} < T < 400^\circ\text{C}$, based on geochemical analysis. The issue of the temperature is crucial and strongly debated. Based on lutetium thermometry Holliger and Devillers (1981) estimated that $T = 260^\circ\text{C}$ for RZ2 and $T = 280^\circ\text{C}$ for RZ3 while Onegin (2012) concluded that $T = 182 \pm 80^\circ\text{C}$ for RZ3. The temperature of RZ10 is estimated to $380^\circ\text{C}$ (Hidaka and Holliger 1998) while the reanalysis by Davis et al. (2014) gave $T = (100 + 30)^\circ\text{C}$. The advantage of the Maxwell distribution assumption is that it avoids to rely on a particular model of the Oklo reactor since the spectrum is determined solely by the temperature.

It was then noted (Lamoreaux and Togerson 2004; Petrov et al. 2006) that above an energy of several eV, the neutron spectrum shifted to a $1/E$ tail because of the absorption of neutrons in uranium resonances. Thus, the distribution was adjusted to include an epithermal distribution

$$n(v) = (1 - f)n_{\text{th}}(v, T) + fn_{\text{epi}}(v),$$

with $n_{\text{epi}} = v_c^2/v^2$ for $v > v_c$ and vanishing otherwise. $v_c$ is a cut-off velocity that
also needs to be specified. The effective cross-section can then be parameterized (Gould et al. 2006) as

$$ \hat{\sigma} = g(T)\sigma_0 + r_0I, \qquad (163) $$

where $g(T)$ is a measure of the departure of $\sigma$ from the $1/v$ behavior, $I$ is related to the resonance integral of the cross-section and $r_0$ is the Oklo reactor spectral index. It characterizes the contribution of the epithermal neutrons to the cross-section. Among the unknown parameters, the most uncertain is probably the amount of water present at the time of the reaction. Gould et al. (2006) chose to adjust it so that $r_0$ matches the experimental values.

These hypothesis on the neutron spectrum and on the temperature, as well as the constraint on the shift of the resonance energy, are summarized in Table 10. Many analyses (Fujii et al. 2000; Petrov et al. 2006; Gould et al. 2006) find two branches for $\Delta E_r = E_r - E_{r0}$, with one (the left branch) indicating a variation of $E_r$. Note that these two branches disappear when the temperature is higher since $\hat{\sigma}(E_r, T)$ is more peaked when $T$ decreases but remain in any analysis at low temperature. This shows the importance of a good determination of the temperature. Note that the analysis of Petrov et al. (2006) indicates that the curves $\hat{\sigma}(T, E_r)$ lie appreciably lower than for a Maxwell distribution and that Fujii et al. (2000) argues that the left branch is hardly compatible with the gadolinium data.

### 5.3.3 From the resonance energy to fundamental constants

The energy of the resonance depends a priori on many constants since the existence of such resonance is mainly the consequence of an almost cancellation between the electromagnetic repulsive force and the strong interaction. But, since no full analytical understanding of the energy levels of heavy nuclei is available, the role of each constant is difficult to disentangle.

In his first analysis, Shlyakhter (1976) stated that for the neutron, the nucleus appears as a potential well with a depth $V_0 \simeq 50$ MeV. He attributed the change of the resonance energy to a modification of the strong interaction coupling constant and concluded that $\Delta g_S/g_S \sim \Delta E_r/V_0$. Then, arguing that the Coulomb force increases the average inter-nuclear distance by about 2.5% for $A \sim 150$, he concluded that $\Delta \alpha_{EM}/\alpha_{EM} \sim 20\Delta g_S/g_S$, leading to $|\alpha_{EM}/\alpha_{EM}| < 10^{-17} \text{yr}^{-1}$, which can be translated to $|\Delta \alpha_{EM}/\alpha_{EM}| < 1.8 \times 10^{-8}$.

The following analysis focused on the fine-structure constant and ignored the strong interaction. Damour and Dyson (1996) related the variation of $E_r$ to the fine-structure constant by taking into account that the radiative capture of the neutron by $^{149}_{62}\text{Sm}$ corresponds to the existence of an excited quantum state of $^{150}_{62}\text{Sm}$ (so that $E_r = E_{150}^* - E_{149} - m_n$) and by assuming that the nuclear energy is independent of $\alpha_{EM}$. It follows that the variation of $\alpha_{EM}$ can be related to the difference of the Coulomb binding energy of these two states. The computation of this latter quantity is difficult and must be related to the mean-square radii of the protons in the isotopes of samarium. In particular this analysis (Damour and Dyson 1996) showed that the
Bethe-Weizäcker formula overestimates by about a factor the 2 the $\alpha_{EM}$-sensitivity to the resonance energy. It follows from this analysis that

$$ \alpha_{EM} \frac{\Delta E_r}{\Delta \alpha_{EM}} \approx -1.1 \text{ MeV}. \qquad (164) $$

The analysis by Jänecke (1972) showed that for the isotopes considered in their analysis, the sensitivity of the ground state energy is accurate at less than 3%.

The sensitivity (164) implies that the constraint on $\Delta E_r$ translates to

$$ -0.9 \times 10^{-7} < \Delta\alpha_{EM}/\alpha_{EM} < 1.2 \times 10^{-7} \qquad (165) $$

at $2\sigma$ level, corresponding to the range $-6.7 \times 10^{-17} \text{yr}^{-1} < \alpha_{EM}/\alpha_{EM} < 5.0 \times 10^{-17} \text{yr}^{-1}$ if $\alpha_{EM}$ is assumed constant. This tight constraint arises from the large amplification between the resonance energy ($\sim 0.1 \text{ eV}$) and the sensitivity ($\sim 1 \text{ MeV}$). The re-analysis of these data together with those of Fujii et al. (2000) including gadolinium found the favored result $\alpha_{EM}/\alpha_{EM} = (-0.2 \pm 0.8) \times 10^{-17} \text{yr}^{-1}$, which corresponds to

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-0.36 \pm 1.44) \times 10^{-8} \qquad (166) $$

and the other branch (indicating a variation; see Table 10) leads to $\alpha_{EM}/\alpha_{EM} = (4.9 \pm 0.4) \times 10^{-17} \text{yr}^{-1}$. This non-zero result cannot be eliminated.

The more recent analysis, based on a modification of the neutron spectrum led respectively to (Petrov et al. 2006)

$$ \Delta\alpha_{EM}/\alpha_{EM} = (3.85 \pm 5.65) \times 10^{-8} \qquad (167) $$

and (Gould et al. 2006)

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-0.65 \pm 1.75) \times 10^{-8}, \qquad (168) $$

at a 95% C.L., both using the formalism of Damour and Dyson (1996).

Olive et al. (2002), inspired by grand unification models, reconsider the analysis of Damour and Dyson (1996) by letting all gauge and Yukawa couplings vary. Working within the Fermi gas model, the over-riding scale dependence of the terms, which determine the binding energy of the heavy nuclei was derived. Parameter- izing the mass of the hadrons as $m_i \propto \Lambda_{QCD}(1 + \kappa_i m_q / \Lambda_{QCD} + \dots)$, they estimated that the nuclear Hamiltonian was proportional to $m_q / \Lambda_{QCD}$ at lowest order, which allows them to conclude that the energy of the resonance is related to the quark mass by

$$ \frac{\Delta E_r}{E_r} \sim (2.5 - 10) \times 10^{17} \Delta \ln \left( \frac{m_q}{\Lambda_{QCD}} \right). \qquad (169) $$

Using the constraint (164), they first deduced that
$$ \left| \Delta \ln \left( \frac{m_q}{\Lambda_{QCD}} \right) \right| < (1-4) \times 10^{-8}. $$

Then, assuming that $\alpha_{EM} \propto m_q^{50}$ on the basis of grand unification (see Sect. 8.4 for details), they concluded that

$$ |\Delta\alpha_{EM}/\alpha_{EM}| < (2-8) \times 10^{-10}. \qquad (170) $$

Similarly, Flambaum and Shuryak (2002), Sisterna and Vucetich (1990), Flambaum and Wiringa (2009) related the variation of the resonance energy to the quark mass. Their first estimate (Flambaum and Shuryak 2002) assumed that it is related to the pion mass, $m_\pi$, and that the main variation arises from the variation of the radius $R \sim 5\text{fm} + 1/m_\pi$ of the nuclear potential well of depth $V_0$, so that

$$ \delta E_r \sim -2V_0 \frac{\delta R}{R} \sim 3 \times 10^8 \frac{\delta m_\pi}{m_\pi}, $$

assuming that $R \simeq 1.2A^{1/3}r_0$, $r_0$ being the inter-nucleon distance. Davis et al. (2014), Davis (2015) proposed the natural parameterisation

$$ \Delta E_r = a \Delta \frac{X_q}{X_q} + b \frac{Z^2}{A^{4/3}} \frac{\Delta \alpha_{EM}}{\alpha_{EM}} \qquad (171) $$

with $|b| \sim 0.5$ MeV within the factor 2 (see however Davis and Hamdan 2015) and the more uncertain parameter $a$ ranging from 10 MeV (Flambaum and Wiringa 2009) to -40 MeV (Davis 2015). This latter result would lead to the constraint on the variation of the quark mass

$$ |\Delta m_q/m_q| < 10^{-9}. \qquad (172) $$

Then, Sisterna and Vucetich (1990) described the nuclear potential by a Walecka model, which keeps only the $\sigma$ (scalar) and $\omega$ (vector) exchanges in the effective nuclear force. Their masses was related to the mass $m_s$ of the strange quark to get $m_\sigma \propto m_s^{0.54}$ and $m_\omega \propto m_s^{0.15}$. It follows that the variation of the potential well can be related to the variation of $m_\sigma$ and $m_\omega$ and thus on $m_q$ by $V \propto m_q^{-3.5}$. The constraint (164) then implies

$$ \left| \Delta \ln \left( \frac{m_s}{\Lambda_{QCD}} \right) \right| < 1.2 \times 10^{-10}. $$

By extrapolating from light nuclei where the N-body calculations can be performed more accurately, it was concluded (Flambaum and Shuryak 2003) that the resonance energy scales as $\Delta E_r \simeq 10(\Delta \ln X_q - 0.1\Delta \ln \alpha_{EM})$, so that the constraint from (Petrov et al. 2006) would imply that $\Delta \ln(X_q/\alpha_{EM}^{0.1}) < 7 \times 10^{-9}$.
### 5.3.4 Conclusion

The various constraints obtained from the analysis of the Oklo phenomena are summarized in Table 11 showing that typically, one shall keep in mind the constraints

$$|\Delta\alpha_{EM}/\alpha_{EM}| \lesssim 10^{-8}.$$

These last results illustrate that a detailed theoretical analysis and quantitative estimates of the nuclear physics (and QCD) aspects of the resonance shift still remain to be carried out. In particular, the interface between the perturbative QCD description and the description in term of hadron is not fully understand: we do not know the exact dependence of hadronic masses and coupling constant on $\Lambda_{QCD}$ and quark masses. The second problem concerns modeling nuclear forces in terms of the hadronic parameters.

At present, the Oklo data, while being stringent and consistent with no variation, have to be considered carefully. While a better understanding of nuclear physics is necessary to understand the full constant-dependence, the data themselves require more insight, particularly to understand the existence of the left-branch.

## 5.4 Meteorite dating

Long-lived $\alpha$- or $\beta$-decay isotopes may be sensitive probes of the variation of fundamental constants on geological times ranging typically to the age of the Solar system, $t \sim (4-5)$ Gyr, corresponding to a mean redshift of $z \sim 0.43$. Interestingly, it can be compared with the shallow universe quasar constraints. This method was initially pointed out by Wilkinson (1958) and then revived by Dyson (1972). The main idea is to extract the $\alpha_{EM}$-dependence of the decay rate and to use geological samples to bound its time variation.

The sensitivity of the decay rate of a nucleus to a change of the fine-structure constant is defined, in a similar way as for atomic clocks [Eq. (128)], as

<table><caption>**Table 11** Summary of the main constraints on the variation of $\alpha_{EM}$, assuming all other constants remains fixed, from the Oklo phenomena</caption><thead><tr><th>Ore</th><th>neutron spectrum</th><th>Δα<sub>EM</sub>/α<sub>EM</sub>(10<sup>-8</sup>)</th><th>Ref</th></tr></thead><tbody><tr><td>?</td><td>Maxwell</td><td>(0 ± 1.8)</td><td>Shlyakhter (1976)</td></tr><tr><td>RZ2 (15)</td><td>Maxwell</td><td>(1.5 ± 10.5)</td><td>Damour and Dyson (1996)</td></tr><tr><td>RZ10,13</td><td>Maxwell</td><td>(-0.36 ± 1.44)</td><td>Fujii et al. (2000)</td></tr><tr><td>RZ2</td><td>Maxwell + epithermal</td><td>(3.85 ± 5.65)</td><td>Petrov et al. (2006)</td></tr><tr><td>RZ2,10</td><td>Maxwell + epithermal</td><td>(-0.65 ± 1.75)</td><td>Gould et al. (2006)</td></tr><tr><td>RZ3,5</td><td>Numerical</td><td>(-0.15 ± 0.85)</td><td>Onegin (2012)</td></tr></tbody></table>
$$s_{\alpha} \equiv \frac{\partial \ln \lambda}{\partial \ln \alpha_{\text{EM}}}. \qquad (173)$$

$\lambda$ is a function of the decay energy $Q$. When $Q$ is small, mainly due to an accidental cancellation between different contributions to the nuclear binding energy, the sensitivity $s_{\alpha}$ maybe strongly enhanced. A small variation of the fundamental constants can either stabilize or destabilize certain isotopes so that one can extract bounds on the variation of their lifetime by comparing laboratory data to geo-physical and Solar system probes.

Assume some meteorites containing an isotope $X$ that decays into $Y$ are formed at a time $t_*$.

$$N_X(t) = N_{X*}e^{-\lambda(t-t_*)}, \quad N_Y(t) = N_{X*}[1 - e^{-\lambda(t-t_*)}] + N_{Y*} \quad (174)$$

if one assumes the decay rate constant. If it is varying then these relations have to be replaced by

$$N_X(t) = N_{X*}e^{\int_{t_*}^{t} \lambda(t')dt'}$$

so that the value of $N_X$ today can be interpreted with Eq. (174) but with an effective decay rate of

$$\bar{\lambda} = \frac{1}{t_0 - t_*} \int_{t_*}^{t_0} \lambda(t') dt'. \qquad (175)$$

From a sample of meteorites, we can measure $\{N_X(t_0), N_Y(t_0)\}$ for each meteorite. These two quantities are related by

$$N_Y(t_0) = [e^{\bar{\lambda}(t_0 - t_*)} - 1]N_X(t_0) + N_{Y*},$$

so that the data should lie on a line (since $N_{X*}$ is a priori different for each meteorite), called an “isochron”, the slope of which determines $\bar{\lambda}(t_0 - t_*)$. It follows that meteorites data only provide an *average* measure of the decay rate, which complicates the interpretation (see Fujii and Iwamoto 2005, 2003 for explicit examples). To derive a bound on the variation of the constant we also need a good estimation of $t_0 - t_*$, which can be obtained from the same analysis for an isotope with a small sensitivity $s_{\alpha}$, as well as an accurate laboratory measurement of the decay rate.

### 5.4.1 Long lived $\alpha$-decays

The $\alpha$-decay rate, $\lambda$, of a nucleus $A_ZX$ of charge $Z$ and atomic number $A$,

$$^{A+4}_{Z+2}X \rightarrow ^A_Z X + ^4_2\text{He}, \qquad (176)$$

is governed by the penetration of the Coulomb barrier that can be described by the Gamow theory. It is well approximated by
$$ \lambda \approx \Lambda(\alpha_{\text{EM}}, \nu) \exp\left(-4\pi Z \alpha_{\text{EM}} \frac{c}{\nu}\right), \qquad (177) $$

where $v/c = \sqrt{Q/2m_p c^2}$ is the escape velocity of the $\alpha$ particle. $\Lambda$ is a function that depends slowly on $\alpha_{EM}$ and $Q$. It follows that the sensitivity to the fine-structure constant is

$$ s_{\alpha} \approx -4\pi Z \frac{\alpha_{\text{EM}}}{\sqrt{Q/2m_{\text{p}}}} \left( 1 - \frac{1}{2} \frac{d}{d \ln \alpha_{\text{EM}}} \ln Q \right). \qquad (178) $$

The decay energy is related to the nuclear binding energies $B(A, Z)$ of the different nuclei by

$$ Q = B(A, Z) + B_{\alpha} - B(A + 4, Z + 2) $$

with $B_\alpha = B(4, 2)$. Physically, an increase of $\alpha_{EM}$ induces an increase in the height of the Coulomb barrier at the nuclear surface while the depth of the nuclear potential well below the top remains the same. It follows that $\alpha$-particle escapes with a greater energy but at the same energy below the top of the barrier. Since the barrier becomes thinner at a given energy below its top, the penetrability increases. This computation indeed neglects the effect of a variation of $\alpha_{EM}$ on the nucleus that can be estimated to be dilated by about 1% if $\alpha_{EM}$ increases by 1%.

As a first insight, when focusing on the fine-structure constant, one can estimate $s_\alpha$ by varying only the Coulomb term of the binding energy. Its order of magnitude can be estimated from the Bethe–Weizsäcker formula (28).

Table 12 summarizes the most sensitive isotopes, with the sensitivities derived from a semi-empirical analysis for a spherical nucleus (Olive et al. 2002). They are in good agreement with the ones derived from Eq. (28) (e.g., for $^{238}$U, one would obtain $s_\alpha = 540$ instead of $s_\alpha = 548$).

The sensitivities of all the nuclei of Table 12 are similar, so that the best constraint on the time variation of the fine-structure constant will be given by the nuclei with the smaller $\Delta\lambda/\lambda$.

Wilkinson (1958) considered the most favorable case, that is the decay of $^{238}_{92}\text{U}$ for which $s_\alpha = 548$ (see Table 12). By comparing the geological dating of the Earth by different methods, he concluded that the decay constant $\lambda$ of $^{238}\text{U}$, $^{235}\text{U}$ and $^{232}\text{Th}$

<table><caption>**Table 12** Summary of the main nuclei and their physical properties that have been used in $\alpha$-decay studies</caption><thead><tr><th>Element</th><th>Z</th><th>A</th><th>Lifetime (yr)</th><th>Q (MeV)</th><th>s<sub>α</sub></th></tr></thead><tbody><tr><td>Sm</td><td>62</td><td>147</td><td>1.06 × 10<sup>11</sup></td><td>2.310</td><td>774</td></tr><tr><td>Gd</td><td>64</td><td>152</td><td>1.08 × 10<sup>14</sup></td><td>2.204</td><td>890</td></tr><tr><td>Dy</td><td>66</td><td>154</td><td>3 × 10<sup>6</sup></td><td>2.947</td><td>575</td></tr><tr><td>Pt</td><td>78</td><td>190</td><td>6.5 × 10<sup>11</sup></td><td>3.249</td><td>659</td></tr><tr><td>Th</td><td>90</td><td>232</td><td>1.41 × 10<sup>10</sup></td><td>4.082</td><td>571</td></tr><tr><td>U</td><td>92</td><td>235</td><td>7.04 × 10<sup>8</sup></td><td>4.678</td><td>466</td></tr><tr><td>U</td><td>92</td><td>238</td><td>4.47 × 10<sup>9</sup></td><td>4.270</td><td>548</td></tr></tbody></table>
have not changed by more than a factor 3 or 4 during the last $3 - 4 \times 10^9$ years from which it follows

$$|\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}| < 8 \times 10^{-3}. \qquad (179)$$

It was revised by Dyson (1972) who claimed that the decay rate has not changed by more than 20% during the past $2 \times 10^9$ years, which implies

$$|\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}| < 4 \times 10^{-4}. \qquad (180)$$

Uranium has a short lifetime so that it cannot be used to set constraints on longer time scales. It is also used to calibrate the age of the meteorites. Therefore, Olive et al. (2002) suggested to consider $^{147}$Sm. Assuming that $\Delta\lambda_{147}/\lambda_{147}$ is smaller than the fractional uncertainty of $7.5 \times 10^{-3}$ of its half-life

$$|\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}| \lesssim 10^{-5}. \qquad (181)$$

As for the Oklo phenomena, the effect of other constants has not been investigated in depth. It is clear that at lowest order both $Q$ and $m_p$ scales as $\Lambda_{QCD}$ so that one needs to go beyond such a simple description to determine the dependence in the quark masses. Taking into account the contribution of the quark masses, in the same way as for Eq. (169), it was argued that $\lambda \propto X_q^{300-2000}$, which leads to $|\Delta \ln X_q| \lesssim 10^{-5}$. In a GUT framework, that could lead to a constraint of the order of $|\Delta \ln \alpha_{EM}| \lesssim 2 \times 10^{-7}$.

### 5.4.2 Long lived $\beta$-decays

Dicke (1959) stressed that the comparison of the rubidium-strontium and potassium-argon dating methods to uranium and thorium rates constrain the variation of $\alpha_{EM}$.

As long as long-lived $\beta$-decay isotopes are concerned for which the decay energy $Q$ is small, we can use a non-relativistic approximation for the decay rate

$$\lambda = \Lambda_{\pm} Q^{p_{\pm}} \qquad (182)$$

respectively for $\beta^-$-decay and electron capture. $\Lambda_{\pm}$ are functions that depend smoothly on $\alpha_{EM}$ and which can thus be considered constant, $p_+ = \ell + 3$ and $p_- = 2\ell + 2$ are the degrees of forbiddenness of the transition. For high-$Z$ nuclei with small decay energy $Q$, the exponent $p$ becomes $p = 2 + \sqrt{1 - \alpha_{EM}^2 Z^2}$ and is independent of $\ell$. It follows that the sensitivity to a variation of $\alpha_{EM}$ is

$$s_{\alpha} = p \frac{d \ln Q}{d \ln \alpha_{EM}}. \qquad (183)$$

The second factor can be estimated exactly as for $\alpha$-decay. We note that $\Lambda_{\pm}$ depends on the Fermi constant and on the mass of the electron as $\Lambda_{\pm} \propto G_F^2 m_e^5 Q^p$. This dependence is the same for any $\beta$-decay so that it disappears in the comparison of two dating methods relying on two different $\beta$-decay isotopes, in which case only the dependence on the other constants appear again through the nuclear binding
energy. Note, however, that comparing a $\alpha$- to a $\beta$-decay may lead to interesting constraints.

We refer to Sect. III.A.4 of FVC03 (Uzan 2003b) for earlier constraints derived from rubidium-strontium, potassium-argon and we focus on the rhenium-osmium case,

$$^{187}_{75}\text{Re} \rightarrow ^{187}_{76}\text{Os} + \bar{\nu}_{e} + e^{-} \qquad (184)$$

first considered by Peebles and Dicke (1962). They noted that the very small value of its decay energy $Q = 2.6$ keV makes it a very sensitive probe of the variation of $\alpha_{EM}$. In that case $p \simeq 2.8$ so that $s_{\alpha} \simeq -18,000$; a change of $10^{-2}\%$ of $\alpha_{EM}$ will induce a change in the decay energy of order of the keV, that is of the order of the decay energy itself. Peebles and Dicke (1962) did not have reliable laboratory determination of the decay rate to put any constraint. Dyson (1967) compared the isotopic analysis of molybdenite ores ($\lambda_{187} = (1.6 \pm 0.2) \times 10^{-11} \text{yr}^{-1}$), the isotopic analysis of 14 iron meteorites ($\lambda_{187} = (1.4 \pm 0.3) \times 10^{-11} \text{yr}^{-1}$) and laboratory measurements of the decay rate ($\lambda_{187} = (1.1 \pm 0.1) \times 10^{-11} \text{yr}^{-1}$). Assuming that the variation of the decay energy comes entirely from the variation of $\alpha_{EM}$, he concluded that $|\Delta\alpha_{EM}/\alpha_{EM}| < 9 \times 10^{-4}$ during the past $3 \times 10^{9}$ years. Note that the discrepancy between meteorite and lab data could have been interpreted as a time-variation of $\alpha_{EM}$, but the lab measurement were complicated by many technical issues so that Dyson only considered a conservative upper limit.

The modelization and the computation of $s_{\alpha}$ were improved in Olive et al. (2002), following the same lines as for $\alpha$-decay.

$$\frac{\Delta\lambda_{187}}{\lambda_{187}} = p\frac{\Delta Q}{Q} \approx p\left(\frac{20 \text{ MeV}}{Q}\right)\frac{\Delta\alpha_{EM}}{\alpha_{EM}} \sim -2.2 \times 10^4\frac{\Delta\alpha_{EM}}{\alpha_{EM}}$$

if one considers only the variation of the Coulomb energy in $Q$. A similar analysis (Dent et al. 2008) led to $\Delta \ln \lambda_{187} \simeq 10^4 \Delta \ln[\alpha_{EM}^{-2.2}X_q^{-1.9}(X_d - X_u)^{0.23}X_e^{-0.058}]$.

The dramatic improvement in the meteoric analysis of the Re/Os ratio (Smoliar et al. 1996) led to a re-analysis of the constraints on the fundamental constants. The slope of the isochron was determined with a precision of 0.5%. However, the Re/Os ratio is inferred from iron meteorites the age of which is not determined directly. Models of formation of the Solar system tend to show that iron and angrite meteorites form within the same 5 million years. The age of the latter can be estimated from the $^{207}\text{Pb}-^{208}\text{Pb}$ method, which gives 4.558 Gyr (Lugmair and Galer 1992) so that $\lambda_{187} = (1.666 \pm 0.009) \times 10^{-11} \text{yr}^{-1}$. Thus, we could adopt (Olive et al. 2002)

$$\left|\frac{\Delta\lambda_{187}}{\lambda_{187}}\right| < 5 \times 10^{-3}.$$

However, the meteoritic ages are determined mainly by $^{238}\text{U}$ dating so that effectively we have a constraint on the variation of $\lambda_{187}/\lambda_{238}$. Fortunately, since the sensitivity of $^{238}\text{U}$ is much smaller than the one of the rhenium, it is safe to neglect its effect. Using the recent laboratory measurement (Lindner et al. 1986)
($\lambda_{187} = (-1.639 \pm 0.025) \times 10^{-11} \text{yr}^{-1}$), the variation of the decay rate is not given by the dispersion of the meteoritic measurement, but by comparing to its value today, so that

$$ \left| \frac{\Delta\lambda_{187}}{\lambda_{187}} \right| = -0.016 \pm 0.016. \qquad (185) $$

The analysis of Olive et al. (2004), following the assumption of Olive et al. (2002), deduced that

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-8 \pm 16) \times 10^{-7}, \qquad (186) $$

at a 95% confidence level, on a typical time scale of 5 Gyr (or equivalently a redshift of order $z \sim 0.2$).

As pointed out in Fujii and Iwamoto (2005, 2003), these constraints really represent a bound on the average decay rate $\bar{\lambda}$ since the formation of the meteorites. This implies in particular that the redshift at which one should consider this constraint depends on the specific functional dependence $\lambda(t)$. It was shown that well-designed time dependence for $\lambda$ can obviate this limit, due to the time average.

The direct test of a possible time variation of the decay rate, or equivalently the half-life, of long lived radioisotopes in the laboratory is difficult mostly because of (1) the duration of the experiments and (2) the necessity to control background phenomena on the long-term since most of them (seasonal cosmic-ray flux modulation, Solar wind and activity, Lunar cycles, tides, ground radon level, etc.) enjoy time and seasonal periodicities. For the first time, such a test was achieved by Arnold et al. (2021) based on 6.0195 yr of NEMO-3 data for the double-$\beta$ decay of Molybdenum to Ruthenium, $^{100}\text{Mo} \rightarrow {}^{100}\text{Ru}$. The half-life of $7 \times 10^{18} \text{yr}$ exhibits no time modulation larger than 2.5% in the frequency range $[0.33225, 365.25]\text{yr}^{-1}$. So far, this constraint has not been translated to any bound on the time variation of constants.

### 5.4.3 Conclusions

Meteorites data allow to set constraints on the variation of the fundamental constants, which are comparable to the ones set by the Oklo phenomenon. Similar constraints can also be set from spontaneous fission (see Sect. III.A.3 of FVC03 Uzan 2003b) but this process is less well understood and less sensitive than the $\alpha$- and $\beta$- decay processes.

From an experimental point of view, the main difficulty concerns the dating of the meteorites and the interpretation of the effective decay rate. Recently, the possibility to directly constrain the half-life of long lived radioisotopes has been proposed.

As long as we only consider $\alpha_{\text{EM}}$, the sensitivities can be computed mainly by considering the contribution of the Coulomb energy to the decay energy, that reduces to its contribution to the nuclear energy. However, as for the Oklo
phenomenon, the dependencies in the other constants, $X_q$, $G_F$, $\mu...$, require a nuclear model and remain very model-dependent.

## 5.5 Quasar absorption spectra

Astrophysics relies heavily on the measurements of redshifts, that is the comparison of the laboratory wavelengths of a particular atomic or molecular transition to its observed value in an astrophysical system, in absorption or emission. The cosmic expansion implies an achromatic redshift of all spectra. There are additional sources of redshift due mostly to peculiar velocity and the Doppler effect they induce and the local gravitational potential of the region from which light is emitted through the Einstein effect. Both are achromatic so that they impact any wavelength in the same way. Indeed, if the values of fundamental constants are spacetime dependent, one expects additional chromatic red or blue shift, depending on the detailed structure of the atom or molecule in question. The situation is actually similar to clock comparisons in the laboratory with the main difference that the physical environment of the observed spectra is not controlled in terms of velocity dispersion of the absorbers/emitters, exact chemical composition, existence of cosmic electric/magnetic fields etc. Among all the astrophysical systems quasistellar objects (QSO), aka. “quasars”, absorption spectra provide a powerful probe of the variation of fundamental constants. Absorption lines in intervening clouds along the line of sight of the QSO give access to the spectra of the atoms present in the cloud, that is to paleo-spectra. As for clocks, the sensitivity of the atomic or molecular transitions that can be observed and the possibility to correlate them will define the level of constraint that can be set. The method was first used by Savedoff (1956) who constrained the time variation of the fine-structure constant from the doublet separations seen in galaxy emission spectra. For general introduction to these observations, we refer to Petitjean et al. (2009b), Srianand et al. (2009), Kanekar (2009). Indeed, one cannot use a single transition to tackle down a variation of the fundamental constants, since one should resort on various transitions and look for chromatic effects that can indeed not be reproduced by the expansion of the universe.

### 5.5.1 Elements for a $\alpha_{EM}$ measurement

To achieve such a test, one needs to consider a series of technical practicalities.

#### *Sensitivities of the atomic transitions*

As for clock comparisons, one needs to understand the dependencies of different types of transitions. An extensive work has been achieved in this direction, still progressing with the discovery of new atomic or molecular lines in astronomical observations.

Dzuba et al. (1999a), Dzuba and Flambaum (2000) proposed to use the convenient formulation
$$ \omega = \omega_0 + q \left[ \left( \frac{\alpha_{EM}}{\alpha_{EM}^{(0)}} \right)^2 - 1 \right] + q_2 \left[ \left( \frac{\alpha_{EM}}{\alpha_{EM}^{(0)}} \right)^4 - 1 \right], \quad (187) $$

in order to take into account the dependence of the spectra on $\alpha_{EM}$. $\omega$ is the energy in the rest-frame of the cloud, that is at a redshift $z$, $\omega_0$ is the energy measured today in the laboratory. $q$ and $q_2$ are two coefficients that determine the frequency dependence on a variation of $\alpha_{EM}$ and that arise from the relativistic corrections for the transition under consideration. The coefficient $q$ is typically an order of magnitude larger than $q_2$ so that the possibility to constrain a variation of the fine-structure constant is mainly determined by $q$. These coefficients are obtained as

$$ q = \frac{\omega(\Delta x) - \omega(-\Delta x)}{2\Delta x} $$

by varying the value of $\alpha_{EM}$ in numerical codes; see Murphy et al. (2004) for a list of these sensitivities. They were computed for a large set of transitions, first using a relativistic Hartree-Fock method and then using many-body perturbation theory. We refer to Dzuba et al. (1999a), Berengut et al. (2006), Angstmann et al. (2004) for an extensive discussion of the computational methods and a list of the $q$-coefficients for various transitions relevant for both quasar spectra and atomic clock experiments. The uncertainty in $q$ are typically smaller than $30 \text{ cm}^{-1}$ for Mg, Si, Al and Zn, but much larger for Cr, Fe and Ni due to their more complicated electronic configurations. The accuracy for $\omega_0$ from dedicated laboratory measurements now reach $0.004 \text{ cm}^{-1}$. It is important to stress that the form (187) ensures that errors in the $q$-coefficients cannot lead to a non zero detection of $\Delta\alpha_{EM}$.

## Accuracy of the measurements

The shifts in the absorption lines to be detected are extremely small. For instance a change of $\alpha_{EM}$ of order $10^{-5}$ corresponds a shift of at most 20 mÅ for a redshift of $z \sim 2$, which would corresponds to a shift of order $\sim 0.5 \text{ km/s}$, or to about a third of a pixel at a spectral resolution of $R \sim 40,000$, as achieved with Keck/HIRES or VLT/UVES. Note that since a few years ago uncertainties in laboratory wavelengths provided the dominant part of the error budget of many measurements.

As we shall discuss later, there are several sources of uncertainties that hamper the measurement. In particular, the absorption lines have complex profiles (because they result from the propagation of photons through a highly inhomogeneous medium) that are fitted using a combination of Voigt profiles. Each of these components depends on several parameters including the redshift, the column density and the width of the line (Doppler parameter) to which one now needs to add the constants that are assumed to be varying. These parameters are constrained assuming that the profiles are the same for all transitions, which is indeed a non-trivial assumption for transitions from different species (this was one of the driving motivations to use the transition from a single species and of the SIDAM method). More important, the fit is usually not unique. This is not a problem when the lines are not saturated but it can increase the error on $\alpha_{EM}$ by a factor 2 in the case of
strongly saturated lines (Chand et al. 2005). To finish, knowing the relative isotopic abundances of some species is also important; see Murphy and Berengut (2014) for a recent compilation of the data for $\alpha_{EM}$ measurements.

It follows that not all transitions are sufficiently sensitive and that only few astrophysical systems are clean enough to provide accurate measurements. Roughly, more than one hundred lines of sight allow for $\alpha_{EM}$ measurements with only a fraction of them being ideal. Besides atomic absorption spectra, molecular absorption spectra allow for $\mu$-measurements—see our discussion in Sect. 5.7—and emission spectra are in general less vulnerable to some systematics but less sensitive; see Sect. 5.8.

### *Hunting systematics.*

In order to claim for a variation of $\alpha_{EM}$, one shall eliminate any systematics that could mimic an effect on an absorption spectra. Many systematics have been listed and studied.

(1) Errors in the determination of laboratory wavelengths to which the observations are compared.

(2) While comparing wavelengths from different atoms one has to take into account that they may be located in different regions of the cloud with different velocities and hence different Doppler shifts.

(3) One has to ensure that there is no transition blended by transitions of another system.

(4) The differential isotopic saturation has to be controlled. Usually quasar absorption systems are expected to have lower heavy element abundances. The spatial inhomogeneity of these abundances may also play a role.

(5) Hyperfine splitting can induce a saturation similar to isotopic abundances.

(6) The variation of the velocity of the Earth during the integration of a quasar spectrum can also induce differential Doppler shift.

(7) Atmospheric dispersion across the spectral direction of the spectrograph slit can stretch the spectrum. It was shown that, on average, this can, for low redshift observations, mimic a negative $\Delta\alpha_{EM}/\alpha_{EM}$, while this is no more the case for high redshift observations, hence emphasizing their complementarity.

(8) The presence of a magnetic field will shift the energy levels by Zeeman effect.

(9) Temperature variations during the observation will change the air refractive index in the spectrograph. In particular, flexures in the instrument are dealt with by recording a calibration lamp spectrum before and after the science exposure and the signal-to-noise and stability of the lamp is crucial

(10) Instrumental effects such as variations of the intrinsic instrument profile have to be controlled.

All these effects have been discussed in details; see e.g., Murphy et al. (2001a, 2001c) and one shall keep in mind when comparing them that VLT/UVES
and Keck/HIRES have, for this particular science case, an irreducible systematic of about 3 parts per million (Whitmore and Murphy 2015).

## Calibration

This was complemented by a study on the crucial issue of calibration since any distortion of the wavelength scale could lead to a non-zero value of $\Delta\alpha_{EM}$. The wavelengths scale were calibrated by mean of a thorium-argon emission lamp. The quality of the calibration of the Keck/HIRES spectrograph was discussed in Murphy et al. (2001a, 2007a) and argued to have negligible effects on the measurements. Griest et al. (2010) provided an analysis of is wavelength accuracy of the Keck/ HIRES. An absolute uncertainty of $\Delta z \sim 10^{-5}$, corresponding to $\Delta\lambda \sim 0.02$ Å with daily drift of $\Delta z \sim 5 \times 10^{-6}$ and multiday drift of $\Delta z \sim 2 \times 10^{-5}$, arguing that this level of systematic uncertainty makes it difficult to use the Keck/HIRES to constrain the time variation of $\alpha_{EM}$ (at least for a single system or a small sample since the distortion pattern pertains to the echelle orders as they are recorded on the CDD, that is it is similar from exposure to exposure, the effect on $\Delta\alpha_{EM}/\alpha_{EM}$ for an ensemble of absorbers at different redshifts would be random since the transitions fall in different places with respect to the pattern of the distortion). A similar result has been obtained for the VLT/UVES data (Whitmore et al. 2010). It was concluded that the ThAr lamp calibration of the quasar wavelength scale was distorted with respect to that established from the Solar spectrum—via reflection from asteroids; see Rahmani et al. (2013)—following that spurious velocity shifts were applied to different transitions at different wavelengths, most-likely causing the observed deviations in the determination of $\alpha_{EM}$ (Whitmore and Murphy 2015).

## Effect of the isotopic abundances

As will be detailed below, one concern of the MM method is the isotopic abundances of Mg II that can affect the low-$z$ sample since any changes in the isotopic composition will alter the value of effective rest-wavelengths. This isotopic composition is assumed to be close to terrestrial $^{24}$Mg:$^{25}$Mg:$^{26}$Mg = 79 : 10 : 11. While no direct measurement of $r_{Mg} = (^{26}\text{Mg} + ^{25}\text{Mg}) / ^{24}\text{Mg}$ in QSO absorber was feasible for a long time due to the small separation of the isotopic absorption lines, it was shown (Gay and Lambert 2000), on the basis of molecular absorption lines of MgH that $r_{Mg}$ generally decreases with smaller metallicity. From the absorption spectrum of the quasar HE0001-2340 observed with VLT/UVES, Agafonova et al. (2011) measured the isotopic ratio $^{24}$Mg:$^{25}$Mg:$^{26}$Mg = (19 ± 11) : (22 ± 13) : (59 ± 6) leading to $r_{Mg} = 4$ at $z = 0.45$, hence showing over-abundance of heavy Mg isotopes compared to the Solar system value of $r_{Mg} = 0.3$. In systems at $z = 1.58$ and 1.65 they conclude that $r_{Mg} \lesssim 0.7$ (resp. 2.6). While the first system is thought to be the fragment of the outflow caused by a SNIa of a high-metallicity white-dwarf, the two others are enriched by AGB stars. This shows the high variability of $r_{Mg}$ and its dependence on the stellar history of the absorption system. In standard models it should be near 0 at zero metallicity since type II supernovae are primarily producers of $^{24}$Mg. It was also argued that $^{13}$C is a tracer of $^{25}$Mg and was shown to be low in the case of HE 0515-4414 (Levshakov et al. 2006b).
However, contrary to this trend, Yong et al. (2003) found that $r_{Mg}$ can reach high values for some giant stars in the globular cluster NGC 6752 with metallicity [Fe/H] ~ -1.6. This led Ashenfelter et al. (2004) to propose a chemical evolution model with strongly enhanced population of intermediate (2–8 $M_{\odot}$) stars, which in their asymptotic giant branch phase are the dominant factories for heavy Mg at low metallicities typical of QSO absorption systems, as a possible explanation of the low-z Keck/HIRES observations without any variation of $\alpha_{EM}$. It would require that $r_{Mg}$ reaches 0.62, compared to 0.27 (but then the UVES/VLT constraints would be converted to a detection). Care needs to be taken since the star formation history can be different in each region, even in each absorber, so that one cannot have a priori use the best-fit obtained from the Keck data to the UVES/VLT data. However, such modified nucleosynthetic history will lead to an overproduction of elements such as P, Si, Al, P above current constraints (Fenner et al. 2005), but this later model is not the same as the one of Ashenfelter et al. (2004) that was tuned to avoid these problems. The calculation of isotope shifts in atoms with a few valence electrons was proposed by Berengut et al. (2004) to determine whether differences in isotope abundances in early universe can contribute to the observed anomalies in quasar absorption spectra, with a negative answer.

### 5.5.2 Evolution of the debate

The claim by (Webb et al. 1999; Murphy et al. 2003; King et al. 2012a) that the Keck/HIRES data indicated that the fine structure constant was smaller in the past relaunched the interest in fundamental constants, both from an experimental and theoretical point of view. For many years, data from the VLT and Keck gave discordant results and were at the heart of a lively debate on the hunt for systematics. It mostly emerged for the result by Webb et al. (2011) over about two decades and their $4\sigma$ of a variation of $\alpha_{EM}$ at low redshift ($1<z<4$). Indeed, despite their obvious successes in other fields, spectrographs such as UVES, HARPS or Keck/HIRES were not built with this science case in mind and were far from optimal for it.

Trying to confirm these results was the main motivation for an ESO/UVES Large Program (Molaro et al. 2013; Rahmani et al. 2013; Evans et al. 2014), a dedicated program to test for the variation of fundamental constants with an optimized sample and methodology with about 40 VLT nights of observations over 2010-2013 with a typical resolution of $R \sim 60\,000$ and a signal-to-noise per pixel of order 100, leading to an expected accuracy of order $10^{-6}$ on $\alpha_{EM}$. The survey was unfortunately not optimized to test the dipole hypothesis by Webb et al. (2011) since it included no target towards the North pole of the dipole.

This led to a huge amount of data and, today, the spectroscopic QSO are usually split into two samples (Martins 2017; Aluri et al. 2023).

* The *Archival data set* of Webb et al. (2011) contains 293 archival measurements from the Keck/HIRES and VLT/UVES spectrographs up to redshift of 4.18. These data, with resolution $R \sim 50\,000$, were originally not designed for the study of fundamental constants, by a large number of observers, under a broad
range of observing conditions, and over a time span of almost a decade. This sample was reanalyzed for the study of the spatial variation of $\alpha_{EM}$ Webb et al. (2011)—see Sect. 8—and shall be then taken with caution. Indeed, it has been well-documented that $\alpha_{EM}$ measurements require particularly careful wavelength calibration procedures, which rely on additional data coeval with the QSO observations. Such additional data are not ordinarily taken for standard observations and cannot be obtained a posteriori. Moreover, these two spectrographs are now known to suffer from significant intra-order and long-range distortions (Whitmore and Murphy 2015). Such limitations may be partially mitigated but cannot be fully eliminated. Its global analysis concludes (Webb et al. 2011)

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-2.16 \pm 0.86) \times 10^{-6}, \quad z_{eff} = 1.50. $$

Note however that concerns do exist concerning the existence of some systematics in the data that have not been fully modeled or corrected so far (Whitmore and Murphy 2015; Dumont and Webb 2017; Webb et al. 2025).

* The Dedicated data set contains about 30 measurements designed for the specific analysis of $\alpha_{EM}$ variation, where ancillary data enabled a more robust wavelength calibration procedure, or using more modern spectrographs that do not suffer from the limitations of VLT/UVES or Keck/HIRES. This includes measurements listed in Table 1 of Martins (2017) and more recent ones from the Subaru telescope (Murphy and Cooksey 2017a) and the X-SHOOTER (Wilczynska et al. 2020), HARPS (Milaković et al. 2020) and ESPRESSO (Welsh et al. 2020; Murphy et al. 2022c) spectrographs. The original spectra have resolution $R \sim 50000 - 150000$ (with the upper end of the range due to ESPRESSO), with the exception of the X-SHOOTER data which only has $R \sim 10000$. The latter are the first direct measurements of $\alpha_{EM}$ in the IR part of the electromagnetic spectrum, extending the redshift range up to $z \sim 7.06$ but their sensitivity is only at the level of tens of ppm, so they do not carry significant weight in the statistical analysis. The weighted mean of all the values in each data set. Similarly, we can identify $\alpha_{EM}$ and effective redshift of Table 13 gives

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-0.23 \pm 0.56) \times 10^{-6}, \quad z_{eff} = 1.29. $$

Clearly these two data sets are discrepant; see e.g., Martins et al. (2022a), Martins (2017) for their comparison. The former has a preference for a negative variation at more than two standard deviations, while the latter value is consistent with the null result. The dedicated database sets more stringent measurements but is, for now, smaller than the archival data set so that their constraining power remains comparable (Martins and Vila Miñana 2019).

The Dedicated data set is given in Table 13 and Fig. 15 and described in the text below.
<table><caption>**Table 13** Summary of the dedicated measurements of $\alpha_{EM}$</caption><thead><tr><th>Object</th><th>z<sub>abs</sub></th><th>Δz<sub>EM</sub>/z<sub>EM</sub></th><th>Spectrographs</th><th>Reference</th></tr></thead><tbody><tr><td>3 sources<br/>[0.7 – 1.5]<br/>z<sub>min</sub> = 1.08</td><td></td><td>(4.3 ± 3.4) × 10<sup>-6</sup></td><td>HIRES</td><td>Songaila and Cowie (2014)</td></tr><tr><td rowspan="5">J0120+2133</td><td rowspan="5">0.576</td><td>(–9.12 ± 39.80 ± 3.64<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>(0.73 ± 6.17 ± 1.77<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>(5.47 ± 18.26 ± 4.03<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>(2.60 ± 3.45 ± 2.38<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>(8.36 ± 11.82 ± 2.84<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>J0026-2857</td><td>1.023</td><td>(3.54 ± 8.54 ± 2.38<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES</td><td>Murphy et al. (2016)</td></tr><tr><td>J0058-0041</td><td>1.072</td><td>(–1.35 ± 6.71 ± 2.51<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td rowspan="2">HE0515-4414</td><td rowspan="2">1.15</td><td>(–0.12 ± 1.79) × 10<sup>-5</sup></td><td>UVES</td><td>Molaro et al. (2008b)</td></tr><tr><td>(0.5 ± 2.4) × 10<sup>-5</sup></td><td>HARPS</td><td>Chand et al. (2006)</td></tr><tr><td rowspan="3">HS1549+1919</td><td rowspan="3">1.143</td><td>(–7.49 ± 4.63 ± 3.02<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES/<br/>HIRES/HDS</td><td>Evans et al. (2014)</td></tr><tr><td>(–0.70 ± 6.43 ± 1.55<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES/<br/>HIRES/HDS</td><td>Evans et al. (2014)</td></tr><tr><td>(–6.42 ± 6.52 ± 3.16<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES/<br/>HIRES/HDS</td><td>Evans et al. (2014)</td></tr><tr><td rowspan="3">HE514-4414</td><td rowspan="3">1.158</td><td>(–1.42 ± 0.55 ± 0.65<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES</td><td>Kotuš et al. (2017)</td></tr><tr><td>(–0.27 ± 2.41) × 10<sup>-6</sup></td><td>HARPS</td><td>Milaković et al. (2020)</td></tr><tr><td>(1.3 ± 1.3 ± 0.4<sub>syst</sub>) × 10<sup>-6</sup></td><td>ESPRESSO</td><td>Murphy et al. (2022c)</td></tr><tr><td>J1237+0106</td><td>1.305</td><td>(–4.54 ± 8.08 ± 3.13<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td rowspan="2">J0058-0041</td><td rowspan="2">1.342</td><td>(3.05 ± 3.30 ± 2.13<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td>(5.67 ± 4.19 ± 2.16<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES</td><td>Murphy et al. (2016)</td></tr><tr><td>J0108-0037</td><td>1.371</td><td>(–8.45 ± 5.69 ± 4.64<sub>syst</sub>) × 10<sup>-6</sup></td><td>VLT</td><td>Murphy et al. (2016)</td></tr><tr><td>HE0001-2340</td><td>1.58</td><td>(–1.5 ± 2.6) × 10<sup>-6</sup></td><td>UVES</td><td>Agafonova et al. (2011)</td></tr></tbody></table>
<table><caption>**Table 13** continued</caption><thead><tr><th>Object</th><th>z<sub>abs</sub></th><th>Δz<sub>EM</sub>/z<sub>EM</sub></th><th>Spectrographs</th><th>Reference</th></tr></thead><tbody><tr><td>J1029+1039</td><td>1.622</td><td>(-1.70 ± 9.80 ± 2.47<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td>HE1104-1805</td><td>1.661</td><td>(-4.70 ± 5.30) × 10<sup>-6</sup></td><td>HIRES</td><td>Songaila and Cowie (2014)</td></tr><tr><td>HE2217-2818</td><td>1.692</td><td>(1.3 ± 2.4 ± 1.0<sub>syst</sub>) × 10<sup>-6</sup></td><td>UVES</td><td>Molaro et al. (2013)</td></tr><tr><td>HS1946+7658</td><td>1.738</td><td>(-7.90 ± 6.20) × 10<sup>-6</sup></td><td>HIRES</td><td>Songaila and Cowie (2014)</td></tr><tr><td>J1944+7705</td><td>1.738</td><td>(12.70 ± 16.30 ± 1.47<sub>syst</sub>) × 10<sup>-6</sup></td><td>HDS</td><td>Murphy and Cooksey (2017b)</td></tr><tr><td>Q1101-264</td><td>1.84</td><td>(5.66 ± 2.67) × 10<sup>-6</sup></td><td>UVES</td><td>Levshakov et al. (2007)</td></tr><tr><td>J11032-2645</td><td>1.839</td><td>(3.3 ± 2.9) × 10<sup>-6</sup></td><td>UVES</td><td>Bainbridge and Webb (2017)</td></tr><tr><td>Q2206-1958</td><td>1.921</td><td>(-4.65 ± 6.01 ± 2.24<sub>syst</sub>) × 10<sup>-6</sup></td><td>VLT</td><td>Murphy et al. (2016)</td></tr><tr><td>Q1755+57</td><td>1.971</td><td>(4.72 ± 4.18 ± 2.16<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td>PHL957</td><td>2.309</td><td>(-0.65 ± 6.42 ± 2.26<sub>syst</sub>) × 10<sup>-6</sup></td><td>HIRES</td><td>Murphy et al. (2016)</td></tr><tr><td></td><td></td><td>(-0.20 ± 12.44 ± 3.51<sub>syst</sub>) × 10<sup>-6</sup></td><td>VLT</td><td>Murphy et al. (2016)</td></tr><tr><td>J0035-0918</td><td>2.34</td><td>(-1.2 ± 1.1) × 10<sup>-5</sup></td><td>ESPRESSO</td><td>Welsh et al. (2020)</td></tr><tr><td>J1120+0641</td><td>5.50726</td><td>(7.42 ± 9.60 ± 1.52<sub>syst</sub>) × 10<sup>-5</sup></td><td>X-SHOOTRER</td><td>Wilczynska et al. (2020)</td></tr><tr><td></td><td>5.95074</td><td>(-22.85 ± 17.11 ± 0.32<sub>syst</sub>) × 10<sup>-5</sup></td><td>X-SHOOTRER</td><td>Wilczynska et al. (2020)</td></tr><tr><td></td><td>6.17097</td><td>(-10.16 ± 14.80 ± 0.42<sub>syst</sub>) × 10<sup>-5</sup></td><td>X-SHOOTRER</td><td>Wilczynska et al. (2020)</td></tr><tr><td></td><td>7.05852</td><td>(12.79 ± 48.66 ± 19.74<sub>syst</sub>) × 10<sup>-5</sup></td><td>X-SHOOTRER</td><td>Wilczynska et al. (2020)</td></tr></tbody></table>

Each quasar specifies a line of sight and thus a celestial direction. The redshifts correspond to the various absorption systems on the line of sight leading to independent constraints on z<sub>EM</sub>

### 5.5.3 Different methods to track for the variation of constants

As emphasized, one needs to measure the redshifts from more than one transitions in the same absorption system. The shift between two lines is easier to measure
when the difference between the *q*-coefficients is large, which occurs, e.g., for two levels with large *q* of opposite signs. Many methods were developed to take this into account.

* The **alkali doublet method** (AD) focuses on the fine-structure doublet of alkali atoms. This method completely avoids the assumption of homogeneity because, by construction, the two lines of the doublet must have the same profile. Indeed the AD method avoids the implicit assumption of the MM method that chemical and ionization inhomogeneities are negligible.

* It was then generalized to the **many-multiplet method** (MM), which uses correlations between various transitions in different atoms using the theoretical information that transitions are almost insensitive to a variation of $\alpha_{EM}$. This is the case of Mg II, which can be used as an *anchor*, i.e., a reference point. To obtain strong constraints one can either compare transitions of light atoms with those of heavy atoms (because the $\alpha_{EM}$ dependence of the ground state scales as $Z^2$) or compare *s*–*p* and *d*–*p* transitions in heavy elements (in that case, the relativistic correction will be of opposite signs). This latter effect increases the sensitivity and strengthens the method against systematic errors. However, the results of this method rely on two assumptions: (i) ionization and chemical homogeneity and (ii) isotopic abundance of Mg II close to the terrestrial value. Even though these are reasonable assumptions, one cannot completely rule out systematic biases that they could induce.

* The **single ion differential alpha measurement method** (SIDAM) avoids the influence of small spectral shift due to ionization inhomogeneities within the

<figure><img src="image_21.png" /><figcaption>**Fig. 14** The cosmic expansion induces an achromatic redshift of all wavelengths which distort all emission/absorption spectra. The variation of a fundamental constant $\alpha$ reveals itself through achromatic effects. Depending on the amplitude and sign of the sensitivity $q$, a transition can have a wavelength that is either blue- or red-shifted compared to the cosmological redshift. The idea is then to correlate different lines, in particular with those of very low $q$ that can be used as anchors, which amounts to question if the redshift determined by all transitions is the same or not</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 15** Constraints on the variation of $z_{EM}$ from the analysis of QSO absorption spectra (dedicated data set); see Table 13</figcaption></figure>

absorber and due to possible non-zero offset between different exposures since it relies on different transitions of a single ion in individual exposure.

Most studies are based on *optical techniques* due to the profusion of strong UV transitions that are redshifted into the optical band (this includes AD, MM, SIDAM and it implies that they can be applied only above a given redshift, e.g., Si IV at $z > 1.3$, Fe II $\lambda1608$ at $z > 1$) or on *radio techniques* since radio transitions arise from many different physical effects (hyperfine splitting and in particular H I 21 cm hyperfine transition, molecular rotation, Lambda-doubling, etc). In the latter case, the line frequencies and their comparisons yield constraints on different sets of fundamental constants including $\alpha_{EM}$, $g_p$ and $\mu$. Thus, these techniques are complementary since systematic effects are different in optical and radio regimes. Also the radio techniques offer some advantages: (1) to reach high spectral resolution ($<1$ km/s), alleviating in particular problems with line blending and the use of, e.g., masers allow to reach a frequency calibration better than roughly 10 m/s; (2) in general, the sensitivity of the line position to a variation of a constant is higher; (3) the isotopic lines are observed separately, while in optical there is a blend with possible differential saturations (see, e.g., Combes 2009 for a discussion).
## Alkali doublet method (AD)

The first method used to constrain the time variation of the fine-structure constant relies on fine-structure doublets splitting for which $\Delta v \propto \alpha_{EM}^2 Z^4 R_{\infty}/2n^3$. Hence, $\alpha_{EM}$, $\Delta v/\bar{v} \propto \alpha_{EM}^2$, so that the variation of the fine structure constant at a redshift $z$ can be obtained as

$$ \left( \frac{\Delta \alpha_{EM}}{\alpha_{EM}} \right) (z) = \frac{c_r}{2} \left[ \left( \frac{\Delta \lambda}{\bar{\lambda}} \right)_z / \left( \frac{\Delta \lambda}{\bar{\lambda}} \right)_0 - 1 \right], $$

where $c_r \sim 1$ is a number taking into account the relativistic corrections. This expression is indeed a simple approach since one should, as for atomic clocks, take into account the relativistic corrections more precisely. Using the formulation (187), one can deduce that $c_r = (\delta q + \delta q_2)/(\delta q + 2\delta q_2)$, where the $\delta q$ are the differences between the $q$-coefficients for the doublet transitions.

The AD method has been applied to several species such as, e.g., C IV, N V, O VI, Mg II, Al III, Si II, Si IV. We refer to Sect. III.3 of FVC03 (Uzan 2003b) for a summary of their results (see also Levshakov 2004) and focus on the three most recent analysis, based on the Si IV doublet. In this particular case, $q = 766$ (resp. 362) $cm^{-1}$ and $q_2 = 48$ (resp. -8) $cm^{-1}$ for Si IV $\lambda1393$ (resp. $\lambda1402$) so that $c_r = 0.8914$. The method is based on a $\chi^2$ minimization of multiple components Voigt profile fit to the absorption features in the QSO spectra. In general such a profile depends on three parameters, the column density $N$, the Doppler width and the redshift. It is now extended to include $\Delta\alpha_{EM}/\alpha_{EM}$. The fit is carried out by simultaneously varying these parameters for each component.

Murphy et al. (2001d) analyzed 21 Keck/HIRES Si IV absorption systems toward 8 quasars to obtain the weighted mean

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-0.5 \pm 1.3) \times 10^{-5}, \quad 2.33 < z < 3.08, \qquad (188) $$

with a mean redshift of $z = 2.6$. The S/N ratio of these data is in the range 15–40 per pixel and the spectral resolution is $R \sim 34000$. Chand et al. (2005) analyzed 15 Si IV absorption systems selected from a VLT-UVES sample containing 31 systems (eliminating contaminated, saturated or very broad systems; in particular a lower limit on the column density was fixed so that both lines of the doublets are detected at more than $5\sigma$) to get the weighted mean,

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-0.15 \pm 0.43) \times 10^{-5}, \quad 1.59 < z < 2.92. \qquad (189) $$

The improvement of the constraint arises mainly from a better S/N ratio, of order 60–80 per pixel, and resolution $R \sim 45000$. Note that combining this result with (188) leads to the weighted mean $\Delta\alpha_{EM}/\alpha_{EM} = (-0.04 \pm 0.56) \times 10^{-5}$ in the range $1.59 < z < 3.02$. The analysis (Martínez Fiorenzano et al. 2003) of seven C IV systems and two Si IV systems in the direction of a single quasar, obtained by the VLT-UVES (during the science verification) led to

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-3.09 \pm 8.46) \times 10^{-5}, \quad 1.19 < z < 1.84. \qquad (190) $$

This is less constraining than the two previous analyses, mainly because the $q$-
coefficients are smaller for C IV (see Petitjean and Aracil 2004 for the calibration of the laboratory spectra). One limitation may arise from the isotopic composition. Silicium has three naturally occurring isotopes with terrestrial abundances $^{28}$Si:$^{29}$Si:$^{30}$Si = 92.23 : 4.68 : 3.09 so that each absorption line is a composite of absorption lines from the three isotopes. However, it was shown that this effect of isotopic shifts (Murphy et al. 2001d) is negligible in the case of Si IV.

### Many multiplet method (MM)

A generalisation of the AD method, known as the many-multiplet was proposed by Dzuba et al. (1999b). It relies on the combination of transitions from different species, taking into account the theoretical information that some transitions are fairly unsensitive to a change of the fine-structure constant (e.g., Mg II or Mg I, hence providing good anchors) while others such as Fe II are more sensitive.

### Early Keck/HIRES data analysis.
The MM-method was first applied by Webb et al. (1999) who analyzed one transition of the Mg II doublet and five Fe II transitions from three multiplets. Using 30 absorption systems toward 17 quasars, they obtained

$$ \begin{aligned} \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} &= (-0.17 \pm 0.39) \times 10^{-5}, & 0.6 < z < 1 \\ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} &= (-1.88 \pm 0.53) \times 10^{-5}, & 1 < z < 1.6. \end{aligned} $$

This was indeed the first claim that a constant may have varied during the evolution of the universe. It was later confirmed by (Murphy et al. 2001c; Webb et al. 2001) from a reanalysis of the initial sample and by including new optical QSO data to reach 28 absorption systems with redshift $z = 0.5 - 1.8$ plus 18 damped Lyman-$\alpha$ absorption systems towards 13 QSO plus 21 Si IV absorption systems toward 13 QSO. The analysis used mainly the multiplets of Ni II, Cr II and Zn II and Mg I, Mg I, Al II, Al III and Fe II was also included. The analysis (Murphy et al. 2003) relies on 128 absorption spectra, later updated (Murphy et al. 2004) to include 143 absorption systems. The more robust estimates is the weighted mean

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-0.57 \pm 0.11) \times 10^{-5}, \quad 0.2 < z < 4.2. \quad (191) $$

The resolution for most spectra was $R \sim 45000$ and the S/N per pixel ranges from 4 to 240, with most spectral regions with S/N $\sim 30$.

The low-$z$ ($z < 1.8$) and high-$z$ rely on different ions and transitions with very different $\alpha_{EM}$-dependencies. At low-$z$, the Mg transitions are used as anchors against which the large positive shifts in the Fe II can be measured. At high-$z$, different transitions are fitted (Fe II, S II, Cr II, Ni II, Zn II, Al II, Al III). The two sub-samples respond differently to simple systematic errors due to their different arrangement of q-coefficients in wavelength space. The analysis for each sample gives the weighted mean
$$ \begin{aligned} \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} &= (-0.54 \pm 0.12) \times 10^{-5}, & 0.2 < z < 1.8 \\ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} &= (-0.74 \pm 0.17) \times 10^{-5}, & 1.8 < z < 4.2, \end{aligned} \qquad (192) $$

with respectively 77 and 66 systems.

**VLT/UVES data analysis.** The previous results led another team to check this detection using the UVES spectrograph operating on the VLT. In order to avoid as much systematics as possible, they apply a series of selection criteria (Chand et al. 2004) on the systems used: (1) consider only lines with similar ionization potentials (Mg II, Fe II, Si II and Al II) as they are most likely to originate from similar regions in the cloud; (2) avoid absorption lines contaminated by atmospheric lines; (3) consider only systems with high enough column density to ensure that all the multiplets are detected at more than $5\sigma$; (4) demand that at least one of the anchor lines is not saturated to have a robust measurement of the redshift; (5) reject strongly saturated systems with large velocity spread; (6) keep only systems for which the majority of the components are separated from the neighboring by more than the Doppler shift parameter. The advantage of this choice is to reject most complex or degenerate systems, which could result in uncontrolled systematics effects. The drawback is indeed that the analysis is based on less systems.

Chand et al. (2004) and Srianand et al. (2004) analyzed the observations of 23 systems satisfying these criteria in direction of 18 QSO with a S/N ranging between 50 and 80 per pixel and a resolution $R > 44000$ to conclude that $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-0.06 \pm 0.06) \times 10^{-5}$ for $0.4 < z < 2.3$, hence giving a $3\sigma$ constraint on a variation of $\alpha_{\text{EM}}$. This analysis was challenged by Murphy et al. (2007b, 2008b, 2008c) who claimed that the same data lead to a weighted mean, $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-0.44 \pm 0.16) \times 10^{-5}$. These arguments were responded in Srianand et al. (2007) that revised the VLT/UVES constraint, rejecting two more than $4\sigma$ deviant systems that were claimed to dominate the re-analysis, to conclude that

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (0.01 \pm 0.15) \times 10^{-5}, \quad 0.4 < z < 2.3. \qquad (193) $$

Let us mention that Molaro et al. (2008b) reanalyzed some systems of Chand et al. (2004), Srianand et al. (2004) by means of the SIDAM method (see below) and disagree with some of them, claiming for a problem of calibration. They also claim that the errors quoted in Murphy et al. (2004) are underestimated by a factor 1.5. This debate has greatly motivated the developments of the Large Program and set the stage for better-controlled analysis of $\alpha_{\text{EM}}$ systems.

**Regressional MM.** The MM method was adapted to use a linear regression method Quast et al. (2004). The idea is to measure the redshift $z_i$ deduced from the transition i and plot $z_i$ as a function of the sensitivity coefficient. If $\Delta\alpha_{\text{EM}} \neq 0$ then there should exist a linear relation with a slope proportional to $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}$. On a single absorption system (VLT/UVES), on the basis of Fe II transition, they concluded that
$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-0.4 \pm 1.9 \pm 2.7_{\text{syst}}) \times 10^{-6}, \quad z = 1.15, \qquad (194) $$

compared to $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (0.1 \pm 1.7) \times 10^{-6}$ that is obtained with the standard MM technique on the same data. This is also consistent with the constraint (195) obtained on the same system with the HARPS spectrograph.

*Analysis of the archival data set.* Webb et al. (2011) split the VLT/UVES data at $z = 1.8$ to get $(\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}})_{\text{VLT}; z < 1.8} = (-0.06 \pm 0.16) \times 10^{-5}$, in agreement with the former study (Srianand et al. 2007), while at higher redshifts $(\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}})_{\text{VLT}; z > 1.8} = (+0.61 \pm 0.20) \times 10^{-5}$. This higher-redshift set exhibits a positive variation of $\alpha_{\text{EM}}$ of opposite sign with respect to the previous Keck/HIRES detection (Murphy et al. 2004) that gave $(\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}})_{\text{Keck}; z < 1.8} = (-0.54 \pm 0.12) \times 10^{-5}$ and $(\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}})_{\text{Keck}; z > 1.8} = (-0.74 \pm 0.17) \times 10^{-5}$. It was pointed out that the Keck/HIRES and VLT/UVES observations can be made consistent in the case the fine structure constant is spatially varying (Webb et al. 2011; Berengut et al. 2011; Berengut and Flambaum 2012), as discussed in Sect. 8 since they do not correspond to the same hemisphere.

### 5.5.4 Single ion differential measurement (SIDAM)

This variation (Levshakov et al. 2006a) on the MM method avoids the influence of small spectral shifts due to ionization inhomogeneities within the absorbers as well as to non-zero offsets between different exposures. It was mainly used with Fe II, which provides transitions with positive and negative q-coefficients. Since it relies on a single ion, it is less sensitive to isotopic abundances, and in particular not sensitive to the one of Mg.

The first analysis relies on the QSO HE 0515-4414 (Quast et al. 2004) set the constraint (194). An independent analysis (Molaro et al. 2008b) gave a weighted mean $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-0.12 \pm 1.79) \times 10^{-6}$ for this system at $z = 1.15$ at $1\sigma$. It was also independently studied using the HARPS spectrograph—that has a higher resolution that UVES; $R \sim 112\,000$—mounted on the 3.6 m telescope at La Silla observatory (Chand et al. 2006). Observations based on Fe II with a S/N of about 30–40 per pixel set the constraint

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (0.5 \pm 2.4) \times 10^{-6}, \quad z = 1.15. \qquad (195) $$

The second constraint (Levshakov et al. 2007; Molaro et al. 2008b) was obtained from a system at $z = 1.84$ toward Q 1101-264, $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (5.66 \pm 2.67) \times 10^{-6}$. A potential systematic uncertainty is the relative shift of the wavelength calibration in the blue and the red arms of UVES where the distant Fe lines are recorded simultaneously (see, e.g., Molaro et al. 2008a for a discussion of the systematics).

A long series of studies, mostly motivated by the Large program with the VLT/UVES led to the construction of the dedicated data set. The actual measurements are all gathered in in Table 13 and we shall only give a short description of these works.
From the study on magnesium isotope abundances with the VLT/UVES Agafonova et al. (2011) set the strong constraint $\Delta\alpha_{EM}/\alpha_{EM} = (-1.5 \pm 2.6) \times 10^{-6}$ in a system at $z_{abs} = 1.5864$ towards HE0001-2340 mainly from FeII, SiII, AlII, AlIII, MgI and MgII.

Evans et al. (2014) observed 3 systems at $z_{abs} = 1.143, 1.342$ and $1.802$ in the direction of HS1549+1919 with 3 telescopes (VLT/UVES, Keck/HIRES and Subaru/HDS). The two low-$z$ systems are constrained thanks to FeII, MgI, and MgII while AlII, AlIII and FeII are used for the higher-$z$ system. The constraint for each system is obtained from the combination of 3 measurements Table 13 and the average on the 3 systems gives $\Delta\alpha_{EM}/\alpha_{EM} = (-5.4 \pm 3.3 \pm 1.5_{\text{syst}}) \times 10^{-6}$ at $1\sigma$.

Songaila and Cowie (2014) used the MM method on 3 quasars observed with Keck/HIRES with MgII and FeII lines at low-$z$ and CrII, ZnII, NiII and MnII lines at higher-$z$. 8 systems with $z$ between 0.7 and 1.5 along the 3 lines of sight are averaged while individual constraints at $z_{abs} = 1.661$ and $1.738$ are obtained for two lines of sight. They concluded that MnII, NiII and CrII give the most robust results. Their constraints are gathered in Table 13.

The first result from the Large program with the VLT/UVES (Molaro et al. 2013) is the analysis of an absorption system at $z_{abs} = 1.692$ towards H22217-2818 from 6 lines of AlII, III and FeII that led to $\Delta\alpha_{EM}/\alpha_{EM} = (1.3 \pm 2.4 \pm 1.0_{\text{syst}}) \times 10^{-6}$.

Murphy et al. (2016) analyzed 9 systems in which Zn and CrII are strong enough to constrain $\Delta\alpha_{EM}/\alpha_{EM}$, 3 of which are observed by both Keck/HIRES and VLT/UVES providing 12 independent data point with $z \in [1.0 - 2.4]$. The 11 independent constraints are gathered in Table 13 and the a weighted mean $\Delta\alpha_{EM}/\alpha_{EM} = (1.2 \pm 1.7 \pm 0.9_{\text{syst}}) \times 10^{-6}$ at $1\sigma$.

Murphy and Cooksey (2017b) tackled the issue the long-range distortions of the wavelength calibration thanks to a supercalibration procedure at the Subaru Telescope/HDS. The first line of sight towards J0120+2133 enjoys 6 absorption systems, only three of which give a constraint on $\alpha_{EM}$. Their average gives $\Delta\alpha_{EM}/\alpha_{EM} = (2.53 \pm 2.87 \pm 2.10_{\text{syst}}) \times 10^{-6}$.

From the analysis of OI, AlII, SiII and FeII with VLT/ESPRESSO in a damped Ly$\alpha$ system at $z_{abs} = 2.34$ towards J0035-0918 Welsh et al. (2020) got $\Delta\alpha_{EM}/\alpha_{EM} = (-1.2 \pm 1.1) \times 10^{-6}$.

The system at redshift $z_{abs} = 1.158$ towards the brightest Southern quasar HE514-4414 is probably the most studied absorber for measuring possible cosmological variations of $\alpha_{EM}$ since Quast et al. (2004) at VLT with UVES (Kotuš et al. 2017) and ESPRESSO (Murphy et al. 2022c) and HARPS at ESO 2.6m telescope (Milaković et al. 2020). The latter analysis made 3 advances on (i) the calibration, using both laser frequency comb and ThAr methods, (ii) the modelization of the spectra with artificial intelligence techniques and (iii) the introduction of additional parameters to measure $\alpha_{EM}$ in each absorption component. They also dispatch their 47 measurements into 37 bins of $\delta z = 10^{-4}$ to analyze the effects of the calibration. Their three measurements are summarized in Table 13. Combining their ESPRESSO results with 28 measurements from other spectrographs in which wavelength calibration errors have been mitigated, allowed Murphy et al. (2022c) to conclude $\Delta\alpha_{EM}/\alpha_{EM} = (-0.5 \pm 0.5 \pm 0.4_{\text{syst}}) \times 10^{-6}$ in the
redshift band 0.6 – 2.4. Lee et al. (2023a) examined the impact of blinding procedures applied in the recent analysis of the same data thanks to supercomputer Monte Carlo AI calculations to generate a large number of independently constructed models of the absorption complex. They conclude that to avoid bias, all future measurements must include $\alpha_{EM}$ as a free parameter from the beginning of the modeling process.

Wilczynska et al. (2020) made the first measurements of $\alpha_{EM}$ with a near-IR spectrograph, VLT/X-SHOOTER. Among the 11 absorption systems identified towards J1120+0641, 4 allowed for a measurement of $\alpha_{EM}$ at redshift $z_{abs} = 7.059$ (CIV, SIV, Nv), 6.171 (AlII, SiII, IV, FeII, MgII), 5.951 (FeII), MgII), SiII), and 5.507 (AlII), FeII), MgII), SiII)) among a sample with a total of 323 measurements spanning the redshift range 0.2 to 0.71 on which they applied the MM method with $\alpha_{EM}$ as a free fitting parameter with IA techniques. Data are gathered in Table 13 and the weighted mean is and is $\Delta\alpha_{EM}/\alpha_{EM} = (-2.18 \pm 7.27) \times 10^{-5}$.

### 5.5.5 Summary of the observable constraints on $\alpha_{EM}$

The analysis of QSO spectra to infer constraints on the variation of $\alpha_{EM}$ is difficult. It has witnessed a lot of developments, in particular to understand all the systematics that can bias the analysis and lead to a spurious non-zero detection. The early Keck/HIRES and VLT/UVES observations and the lively debate between their discordant conclusions led to the development of the Large Program at the VLT and the construction of a dedicated data set.

Among the important advances for $\alpha_{EM}$ measurements, we have mentioned—see e.g., Levshakov (2016), Milaković et al. (2022) for prospective discussions—(1) the control of the calibration, and in particular the joint use of laser frequency comb (Steinmetz et al. 2008; Schmidt et al. 2021) and ThAr lamps, the former leading to a wavelength calibration residuals six times smaller than when using the standard ThAr calibration; (2) the use of Artificial Intelligence and the use of genetic algorithm to develop automated pipelines (Bainbridge and Webb 2017; Lee et al. 2021; Webb et al. 2025) and (3) the use of additional model parameters to measure $\alpha_{EM}$ in each individual absorption component in order to reduce the statistical uncertainty.

The search for a better resolution is being investigated in many directions. With the a resolution of $R \sim 40\,000$, the line wavelengths can be determined with an accuracy of $\sigma_{\lambda} \sim 1$ mÅ. This implies an accuracy of the order of $10^{-5}$ on $\Delta\alpha_{EM}/\alpha_{EM}$ for lines with typical q-coefficients. This limit can indeed be pushed to $10^{-6}$ when more transitions or systems are used together. Any improvement is then related to the possibility to measure line positions with higher accuracy. This can be done by increasing $R$ up to the point at which the narrowest lines in the absorption systems are resolved. The Bohlin formula (Bohlin et al. 1983) gives the estimates

$$ \sigma_{\lambda} \sim \Delta\lambda_{\text{pix}} \left( \frac{\Delta\lambda_{\text{pix}}}{W_{\text{obs}}} \right) \frac{1}{\sqrt{N_e}} \left( \frac{M^{3/2}}{\sqrt{12}} \right), $$

where $\Delta\lambda_{pix}$ is the pixel size, $W_{obs}$ is the observed equivalent width, $N_e$ is the mean