One-electron atoms in screened modified gravity

As we are interested in the, admittedly small, gravitational effects on atoms, we should wonder if the background spacetime causes an appreciable perturbation. For a point-particle nucleus traversing the worldline $\Gamma$ in a background spacetime $\bar{g}_{\mu\nu}$, there always exists adapted coordinates $(t,x^{i})$ such that the rest frame of the nucleus can be Taylor expanded to give

where all derivatives are evaluated at the spatial origin $x^{i}=0$ but still generally remain functions of time $t$. While the choice of coordinates $(t,x^{i})$ is by no means unique, the Fermi coordinates are especially useful Manasse and Misner (1963); Poisson et al. (2011). These preserve the notion of proper time $t$ along the entire extent of the worldline $\Gamma$ and the notion of proper distances $x^{i}$ in its convex normal neighborhood. At the order prescribed, two types of terms appear: terms linear in the background Riemann tensor $\bar{R}_{\mu\nu\rho\sigma}$, which describe tidal forces, and terms up to quadratic order in the acceleration $a^{\mu}$ of $\Gamma$. The interested reader can find explicit expressions for the metric components in Poisson et al. (2011).

The effect of the Riemann tensor terms are studied in Parker (1980), with the result that the leading contribution to the Hamiltonian is

where $\beta$ is a Dirac matrix as defined in Sec. IV. Making an order of magnitude estimate and substituting $\bar{R}_{0i0j}\sim\mathcal{D}^{-2}$ with some curvature length scale $\mathcal{D}$, this perturbation is comparable to fine-structure splitting only when $\mathcal{D}\lesssim Z^{-3}\,10^{-4}\,\text{cm}$, where $Z$ is the atomic number. Galactic (stellar mass) black holes, assumed to be some of the most strongly curved regions of spacetime in our Universe, have $\mathcal{D}\sim 10^{5}\,\text{cm}$ near its event horizon. On these length scales, the perturbation would give rise to an energy shift $\sim 10^{-24}\,\text{eV}$; on Earth it would be $10^{-38}\,\text{eV}$. This is unobservable in both cases.

Similarly, the effect of an accelerating worldline is studied in Dai (2016), with the leading contribution being

If we again wish for this to be comparable to fine-structure splitting, this requires $a\gtrsim 10^{18}\,g_{\oplus}$, where $g_{\oplus}$ is the Earth’s surface gravity. On Earth, this perturbation would give rise to energy shifts $\sim 10^{-21}\,\text{eV}$. While consistently a larger effect than tidal forces, this is also unobservable. Consequently, contributions from a curved background can be safely ignored in what follows.

In the rest frame of the nucleus, a background chameleon can be Taylor expanded to give $\phi^{(0)}=\bar{\phi}+\bar{\phi}_{i}x^{i}+\mathcal{O}(r^{2})$. The constant $\bar{\phi}$ is the chameleon’s vacuum expectation value (vev) in the neighborhood of the atom, and $\bar{\phi}_{i}\equiv\partial_{i}\bar{\phi}|_{x=0}$ is a background gradient. Overlaying this background is a local profile $\delta\phi=\phi-\phi^{(0)}$ due to the presence of the nucleus. Linearizing Eq. (5b) about $\phi^{(0)}$, and considering only static solutions, we obtain

where $\nabla^{2}\equiv\delta^{ij}\partial_{i}\partial_{j}$ is the flat-space Laplacian, and overbars denote a quantity evaluated at $\bar{\phi}$, e.g., $\bar{\Omega}\equiv\Omega(\bar{\phi})$. This equation retains only the two dominant sources for $\delta\phi$, the mass $m_{N}$ and the Coulomb field ${\bf E}_{(0)}$ of the nucleus. Several subleading terms have been safely discarded:

1. (1)

   On the lhs, we ignore the kinetic term $\bar{\phi}^{i}\partial_{i}\delta\phi$ which describes a local-background chameleon interaction. If the background changes on a length scale $\mathcal{D}$, and the Bohr radius is $a_{0}$, then this term is suppressed by at least a factor $a_{0}/\mathcal{D}\ll 1$ relative to $\nabla^{2}\delta\phi$.

2. (2)

   On the rhs, we have ignored the magnetic moment of the nucleus. The resulting magnetic field ${\bf B}_{(0)}$ is always several orders of magnitude weaker than the Coulomb field. Note that excluding the magnetic contribution here does not preclude us from studying the leading chameleon effect on hyperfine structure in Sec. V.4.

3. (3)

   We have also ignored any large-scale, background electromagnetic fields. Relaxing this assumption would give rise to two types of terms. The first are purely background terms $\sim{\bf E}^{2}_{\text{(bg)}}$, ${\bf B}^{2}_{\text{(bg)}}$ which source the background chameleon $\phi^{(0)}$ ¹¹1Note that electromagnetic waves are special in that their contribution $F^{2}=2({\bf B}^{2}-{\bf E}^{2})$ vanishes, so they have no effect altogether on the background chameleon., having no bearing on the form of $\delta\phi$. The second type are cross terms proportional to ${\bf E}_{(0)}\cdot{\bf E}_{\text{(bg)}}$ and ${\bf B}_{(0)}\cdot{\bf B}_{\text{(bg)}}$. The conclusions of this paper hold provided $|{\bf E}_{\text{(bg)}}|,|{\bf B}_{\text{(bg)}}|\ll|{\bf E}_{(0)}|$ in the vicinity of the nucleus, which is typical for environments that host atoms, rather than ions.

4. (4)

   Finally, we have also ignored the term $m_{\phi}^{2}\delta\phi$ that accounts for the chameleon’s effective mass in this environment,

   $$m^{2}_{\phi}=\frac{1}{M_{\text{Pl}}^{2}}\frac{\text{d}}{\text{d}\phi}\left.\left(\frac{V_{,\phi}}{\Omega^{2}}-\beta_{m}\Omega^{2}T^{(0)}\right)\right|_{\bar{\phi}},$$

   (12)

   where $T^{(0)}$ is the background matter density. This expression omits background electromagnetic fields, whose effect is typically small compared with the matter density. We can neglect this term if the local chameleon’s Compton wavelength $m_{\phi}^{-1}$ is much larger than the Bohr radius, that is, $m_{\phi}\ll\zeta m_{e}\approx Z\,(3.7\,\text{keV})$. This is easily satisfied by most, if not all, screening mechanisms of physical interest. Of course, our results equally apply to unscreened scalar fields with mass below this upper limit.

Taking the electric field to be $|{\bf E}_{(0)}|=Ze/4\pi\bar{\varepsilon}r^{2}$ (see Sec. III.4), the solution is

In what follows, one finds that $\bar{\Omega}^{2}$ always accompanies a factor of $(8\pi M_{\text{Pl}}^{2})^{-1}$, and $\bar{\varepsilon}^{-1}$ always accompanies the fine-structure constant $\alpha$. The reason for this is clear: The effect of $\bar{\Omega}$ is to induce an effective gravitational constant, while $\bar{\varepsilon}$ induces an effective fine-structure constant. For brevity, we will henceforth write

We will also drop the overbars on $\beta_{m}$ and $\beta_{\gamma}$.

This local chameleon field and the mass of the nucleus go on to source a metric perturbation $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$. Linearizing Eq. (5a), we find

The stress tensor of the nucleus is $\delta T_{\mu\nu}=m_{N}\delta^{(3)}({\bf r})\delta_{\mu}^{0}\delta_{\nu}^{0}$, and the linearized Einstein tensor is

where $\hat{h}_{\mu\nu}$ is the trace-reversed metric perturbation. For similar reasons, we have ignored the term that is of the order $\sim m_{\phi}^{2}\delta\phi\,\eta_{\mu\nu}$ on the rhs of Eq. (15).

It is easy to verify that this has the solution

where the first term is the familiar weak-field metric with Newtonian potential $\Phi_{\text{N}}=-Gm_{N}/r$, and the second is a fifth force potential $\Phi_{\text{S}}=\beta_{m}\delta\phi$. Defining $\Phi_{\pm}=\Phi_{\text{N}}\pm\Phi_{\text{S}}$, the metric can be written as

The nonvanishing components of the Christoffel symbols are

It will later prove useful to express the metric in terms of the vierbeins $\{e_{\mu}^{\hat{a}}\}$, $g_{\mu\nu}=\eta_{\hat{a}\hat{b}}e^{\hat{a}}_{\mu}e^{\hat{b}}_{\nu}$, where hatted Roman indices $\hat{a},\hat{b}$ denote local Lorentz indices. When the distinction is unnecessary, the hats will be dropped. Their dual vector fields are defined as $e^{\mu}_{\hat{a}}=\eta_{\hat{a}\hat{b}}g^{\mu\nu}e^{\hat{b}}_{\nu}$. For the weak-field metric in Eq. (18), these are

Also useful are expressions for the Ricci tensor, whose non-vanishing components are

In the Lorenz gauge $\nabla^{\mu}A_{\mu}=0$, Eq. (5c) can be rewritten as

which must be solved on the weak-field metric to consistently keep all terms linear in $\phi$. Ignoring any large-scale, background electromagnetic fields as before, we expand $A_{\mu}=A_{\mu}^{(0)}+\delta A_{\mu}$, where $A_{\mu}^{(0)}$ describes the bare electric and magnetic fields of the nucleus,

sourced by its charge and magnetic moment, respectively. The constant $\zeta=Z\alpha/\bar{\varepsilon}$ is as defined in Sec. III.2, $I^{j}$ is the spin operator of the nucleus, and $\gamma=g\alpha/2\bar{\varepsilon}m_{p}$ is proportional to the nuclear gyromagnetic ratio ²²2Precisely, the nuclear gyromagnetic ratio multiplied by $e/4\pi\bar{\varepsilon}$..

Linearizing Eq. (22), we obtain

These equations admit the natural interpretation that gravitational effects generate secondary charges and currents, which then source corrections to the bare electromagnetic fields. The appropriate boundary condition for $\delta A_{\mu}$ is, therefore, that it vanishes in the absence of these effects. Said differently, we demand that the complementary functions be zero. For the zeroth-order gauge fields as given in Eq. (23), the solutions are

The Lorenz gauge at first order is

and we have verified that our solutions satisfy this condition.

Having included the effect of a chameleon-photon interaction sourcing corrections to the electromagnetic fields, our Hamiltonian differs from that in Brax and Burrage (2011). It is therefore worthwhile beginning our discussion by revisiting some of their analyses. The chameleon fine-structure corrections are

We have ignored the background gradient $\bar{\phi}_{i}$ for now, as this is constrained by torsion experiments Khoury and Weltman (2004a) to be negligible on Earth. Its effect in unscreened environments is considered in Sec. V.5.

The second term in Eq. (34) is, roughly speaking, a coupling between the electron spin and the chameleon. For spherically symmetric profiles $\Phi_{\text{S}}$ as in Eq. (13), this term has a zero expectation value because of an exact cancellation between components of the electron wave function Adkins (2008) (further details are also given in the Appendix). However, this is no longer true if we relax any of the first three assumptions made in Sec. III.2, as these will generically break spherical symmetry. We do not consider such complications in this paper since we expect them to be subleading. Nevertheless, this term is interesting because it does not appear altogether if we work in the Einstein frame. This suggests an inequivalence between the two frames, and is discussed in Sec. V.6.

For now, the remaining terms of interest give

where $\mathcal{A}_{q}=\langle r^{q}\rangle$ and $\mathcal{B}_{q}=\langle\beta r^{q}\rangle$ are radial expectation values. Explicit expressions are given in Suslov (2009) and are reproduced in the Appendix. Note that only the $\mathcal{B}$ terms were taken into account in Brax and Burrage (2011).

Let us define the additive correction to the energy of a transition line as $\delta E(\psi-\psi^{\prime})=\langle\delta H\rangle(\psi)-\langle\delta H\rangle(\psi^{\prime})$, for two states $\psi,\psi^{\prime}$. Then the Ly $\alpha$ line has an extra correction $\delta E(2S_{1/2}-1S_{1/2})\approx(10^{-38}\,\text{eV})\beta_{m}^{2}+\text{(smaller terms)}$ due to chameleon effects (see Table 1 for the full expression). As was found in Brax and Burrage (2011), the dominant term, assuming $\beta_{\gamma}$ is not very much larger than $\beta_{m}$, comes from the mass of the nucleus sourcing a chameleon profile, which then couples to the electron mass via the metric. This is encapsulated in the $\mathcal{B}_{-1}$ term. Also worth mentioning is the $\mathcal{A}_{-3}$ term, which can be thought of as a chameleon-mediated backreaction of the electric field on itself. Containing a factor of $\beta_{\gamma}^{2}$, this term allows us to set an upper bound on $\beta_{\gamma}$, whereas previously only an upper bound on the product $\beta_{m}\beta_{\gamma}$ was possible.

We constrain these parameters as follows: Also provided in Table 1 are the standard theoretical uncertainties for the energy level of each state $\sigma(\psi)_{\text{th}}$ as predicted by QED, and the standard uncertainty associated with the experimental measurement $\sigma(\psi-\psi^{\prime})_{\text{exp}}$. For a conservative bound, we simply add these together to form a total standard uncertainty $\sigma(\psi-\psi^{\prime})_{\text{tot}}$. For the Ly $\alpha$ line, we therefore require that the chameleon effects be smaller than $\sigma(2S_{1/2}-1S_{1/2})_{\text{tot}}=1.1\times 10^{-11}\,\text{eV}$, which yields (see also Fig. 1)

at the 68% confidence level. We have repeated this exercise for all hydrogen lines in Table XVI of Mohr et al. (2016) with a relative standard uncertainty better than $10^{-10}$, concluding that the tightest bounds from hydrogen indeed come from the Ly $\alpha$ line. (We will later see in Sec. V.4 that the 21 cm line has little constraining power.) It is worth commenting that our bound on $\beta_{m}$ is marginally better than what was found in Brax and Burrage (2011) purely because we have used more recent values of $\sigma(\psi)$.

Limited by the theoretical uncertainty $\sigma(\psi)_{\text{th}}$, our updated constraints remain far from competitive and are unlikely to improve in the foreseeable future. The best bounds to date come from atom interferometry experiments Burrage et al. (2015); *AI2, which are able to constrain $\log_{10}\beta_{m}\lesssim 5$ ⁵⁵5All constraints, apart from those derived from atomic precision tests, are given at the 95% confidence level.. A different region of parameter space around $\beta_{m}\approx 1$ is also excluded from torsion experiments Adelberger et al. (2007), although these apply only for a limited range of effective masses $m_{\phi}$ and can be evaded if the screening mechanism is adjusted (see Burrage and Sakstein (2016) for a review). Our constraint on $\beta_{\gamma}$ is similarly poor. The tightest bound again arises from torsion experiments Adelberger et al. (2007), yielding $\log_{10}\beta_{\gamma}\lesssim 3$, but with the same caveats. A more universal bound from collider experiments Brax et al. (2009) gives $\log_{\text{10}}\beta_{\gamma}\lesssim 13.5$. The GammeV–CHASE experiment Steffen et al. (2010) lowers this to $\log_{\text{10}}\beta_{\gamma}\lesssim 11$ for chameleons with effective mass $m_{\phi}\lesssim 1\,\text{meV}$.

At this point, we must clarify a caveat to our calculations in Table 1. The $\mathcal{A}_{-3}$ integral formally diverges for $S_{1/2}$ and $P_{1/2}$ states, because their wave functions are nonzero at the origin, and a chameleon-induced correction to the Coulomb potential $\propto r^{-3}$ is sufficiently steep. This is unphysical, and we naturally expect the finite size of the nucleus to regularize this divergence. For radii smaller than the nuclear charge radius $r_{N}$, the bare Coulomb potential $\propto r^{-1}$ is replaced by $A_{0}^{(0)}=-\zeta F(r)$, where $F$ is some form factor due to the nuclear charge distribution. As a crude example, $F=(2r_{N})^{-1}(3-r^{2}/r_{N}^{2})$ for a uniformly charged sphere. In principle, this is calculable for any given model charge distribution, but we will remain agnostic about the exact form of $F$ in our argument. The delta function approximating the mass density of the nucleus must also be replaced by some distribution for $r\leq r_{N}$.

As a result, the $r^{q}$ ($q<0$) dependence of each of the terms in the perturbation Hamiltonian becomes

where $\mathcal{F}(r)$ is the corresponding form factor, which is generally different for each term in $\delta H$. Since the mass and charge distributions of the nucleus are now regular near the origin, so is $\mathcal{F}(r)$. Further assuming that $r^{q}$ and $r_{N}^{q}\mathcal{F}(r)$, and their first derivatives, match at the boundary $r=r_{N}$, this suggests $\mathcal{F}$ is bounded between $\mathcal{F}(r_{N})=1$ and $\mathcal{F}(0)>1$, but of order unity. At leading order, we can approximate $\mathcal{F}(r)\approx\mathcal{F}(0)$.

Therefore, and only when necessary, we regularize the expectation values $\mathcal{A}_{q}$ using a cutoff radius,

Rather than keep an explicit factor of $\mathcal{F}(0)$ in the first integral, we absorb its unknown value into $r_{\text{cut}}$, which we then allow to differ slightly from $r_{N}$. The $q=-3$ integral depends (almost) logarithmically on $r_{\text{cut}}$ (see Fig. 2), suggesting that this term is not sensitive to the detailed nuclear structure and that our approximation captures the physics sufficiently well. We therefore set $r_{\text{cut}}=r_{N}\sim A^{1/3}r_{p}$ in this paper, where $A$ is the mass number and $r_{p}$ is the proton radius.

We further note that all the divergent terms in $\delta H$, once regularized, predict energy shifts that are many orders of magnitude smaller than those predicted by the other, finite terms. So even if one were uneasy about this procedure, it does not discount the predictive power of the finite terms. Given that we believe we have a good understanding of why and how these singularities should be removed, we have found it interesting to include their effects in our discussion.

Both the $\mathcal{A}_{-2}$ and $\mathcal{B}_{-2}$ terms have a similar radial dependence, but they arise from different physics. The former is the mass of the nucleus sourcing a correction to the electric field, which explains the factor of $\zeta m_{N}$. The latter is the electric field sourcing a chameleon profile, which couples to the electron mass via the metric, thus picking up a factor of $Z\zeta m_{e}$. The inclusion of the $\mathcal{A}_{-2}$ term in this work explains why we find a much larger correction to the Lamb shift, on the order of $m_{p}/m_{e}$ for hydrogen.

It is interesting to ask if this larger contribution is able to resolve the proton radius puzzle. Stated simply, the puzzle is a $7\sigma$ discrepancy between measurements of the proton charge radius from experiments using electronic and muonic hydrogen. Electron-proton scattering and measurements of hydrogen and deuterium transition lines give $r_{p}=0.8751(61)\,\text{fm}$ Mohr et al. (2016), whereas a measurement of the Lamb shift in muonic hydrogen Antognini et al. (2013) gives $r_{p}=0.84087(39)\,\text{fm}$. One approach to resolving this puzzle, and the one employed here, is to assume that the radius inferred from electronic hydrogen experiments is accurate. One then has to introduce “new physics” that gives a correction $\delta E(2P_{1/2}-2S_{1/2})(\mu^{-})\approx 300\,\mu\text{eV}$ Carlson (2015) to resolve the discrepancy between the theoretical and experimental values of the muonic Lamb shift.

The solid line in Fig. 1 shows the set of possible values of $\beta_{m}$ and $\beta_{\gamma}$ that allow the chameleon to resolve this puzzle, and is ruled out by constraints placed in Sec. V.1. The dotted line shows the same set of values inferred from the previous analysis in Brax and Burrage (2011). While the ratio of the $\mathcal{A}_{-2}$ to $\mathcal{B}_{-2}$ terms is of order $10^{4}$ in electronic hydrogen, this reduces to $m_{p}/m_{\mu}\sim 10$ in muonic hydrogen, explaining why the solid line is lowered only by an order of magnitude. Many other solutions have been proposed (see Carlson (2015) for a review, and Burgess et al. which introduces no new physics but proposes a change of boundary conditions at small $r$ on the electron wave function), although the puzzle still remains an open problem.

We consider the effects of the nuclear spin in this section. We have neglected a magnetic contribution ${\bf B}_{(0)}^{2}$ as a source for the chameleon profile in Eq. (11), as it is subleading with $\mathcal{O}(\gamma^{2})$, whereas all terms here are linear in $\gamma$. Once again ignoring the background gradient $\bar{\phi}_{i}$, the Hamiltonian is

Both terms have operators of the form $\epsilon_{ijk}\alpha^{i}I^{j}x^{k}r^{-n}$, not unlike the hyperfine structure term in Eq. (32b), and hence can be treated using similar methods. Following the approach in Rose (1961); *RoseAngMom, we find

where $\kappa$ is a quantum number, related to the angular-momentum quantum numbers $j$ and $l$ by Eq. (3); ${\bf F}={\bf I}+{\bf j}$ is the total angular momentum of the system; and $\mathcal{C}_{q}=\langle i\alpha^{r}\beta r^{q}\rangle$ is a radial expectation value, with $\alpha^{r}$ denoting the projection of the Dirac $\alpha$ matrices along the radial direction. For $S_{1/2}$ and $P_{1/2}$ states, both $\mathcal{C}_{-3}$ and $\mathcal{C}_{-4}$ are singular, and we regularize just as we did for $\mathcal{A}_{-3}$ in Sec. V.2.

We find that the ground-state hyperfine transition picks up the correction

We have included a rough uncertainty in the second term because the $\mathcal{C}_{-4}$ integral depends on the cutoff as $\mathcal{C}_{-4}\sim r_{\text{cut}}^{-1}$ (this behavior is briefly explained in the Appendix). It follows that this term is more sensitive to the form factor $\mathcal{F}(r)$. We expect that the uncertainty due to our crude approximation can be at most $r_{\text{cut}}\sim 10^{0\pm 1}r_{p}$, which is reflected in the above equation.

For comparison, the leading hyperfine-structure term [Eq. (32b)] gives an energy splitting $\sim 10^{-6}\,\text{eV}$ in hydrogen. While measurements of this transition have been performed with great precision, achieving relative standard uncertainties at the level of $10^{-12}$ Karshenboim (2005), larger theoretical uncertainties make this transition ill-suited for constraining parameters. This arises from a poor understanding of how the nucleus affects these states. However, even if theoretical predictions were able to match experimental precision, it is easy to check that this still produces weaker bounds on $\beta_{m}$ and $\beta_{\gamma}$ than in Eq. (36).

Our discussion so far has been concerning chameleon effects that might be present on Earth. In unscreened, astrophysical environments, the chameleon vev $\bar{\phi}$ and its gradient $\bar{\phi}_{i}$ can lead to violations of the Einstein equivalence principle (EP).

Recall that the effect of $\bar{\phi}$ is to induce effective gravitational and fine-structure constants [Eq. (14)]. If this has a value $\bar{\phi}_{\oplus}$ on Earth, we normalize $\Omega(\bar{\phi}_{\oplus})=\varepsilon(\bar{\phi}_{\oplus})=1$, such that the bare constants $G_{\textup{N}}$ and $\alpha$ take their usual measured values. Deviations of $\Omega$ and $\varepsilon$ from unity give rise to a fractional change in all energy levels. For nonrelativistic states with $\zeta\ll 1$, the gross-structure energy levels are approximated by the modified Rydberg formula

For $\bar{\varepsilon}$ sufficiently different from unity, this can be an observable effect. However, because this fractional change applies equally to all transition lines, distinguishing it from a cosmological redshift is difficult. In fact, it has long been understood that the spectra of many-electron atoms is far more sensitive to variations in the fine-structure constant. In particular, comparing absorption lines of Fe V and Ni V in the atmospheres of white dwarfs with those measured in the laboratory has been found to be a useful probe of this effect Berengut et al. (2013); *WhiteDwarf2.

We might wonder if the gradient $\bar{\phi}_{i}$ stands a better chance of detection in hydrogen. It leads to a fine-structure correction

where we have aligned the $z$ axis of the atom’s rest frame with $\bar{\phi}_{i}$ in the second equality. This term is analogous to the Stark effect. As $\cos\theta$ is an odd-parity operator, the matrix element $\langle\psi^{\prime}|\cos\theta|\psi\rangle$ vanishes unless the states $\psi,\psi^{\prime}$ have opposite parity. This operator therefore mixes degenerate states with the same principal quantum number $n$ but with differing angular-momentum quantum numbers $\kappa=\pm|\kappa|$, such as states in the $S_{1/2}$ and $P_{1/2}$ levels. The energy splitting and the eigenstates can then be found using standard methods of degenerate perturbation theory. Unfortunately, QED corrections Mohr et al. (2016); Horbatsch and Hessels (2016) much larger than this chameleon effect lift such degeneracies, nullifying any response to the presence of a chameleon gradient. The degeneracy between the $2S_{1/2}$ and $2P_{1/2}$ levels is famously broken by the Lamb shift, for instance.

How large must the background gradient be that its effect is not forbidden by QED corrections? If Eq. (43) is to be comparable to, or greater than, the Lamb shift $\sim 10^{-6}\,\text{eV}$, we require $\beta_{\gamma}|\nabla\bar{\phi}|\gtrsim Z^{-1}\,10^{-4}\,\text{eV}\approx Z^{-1}\,10^{19}\,g_{\oplus},$ where we recall that $g_{\oplus}$ is the surface gravity on Earth. The best candidates for extreme surface gravities are neutron stars, but even they have surface gravities of “only” $\sim 10^{11}\,g_{\oplus}$. But this large surface gravity also implies that a chameleon will be screened by the thin-shell effect Khoury and Weltman (2004a), so its gradient $|\nabla\bar{\phi}|\ll 10^{11}g_{\oplus}$.

The chameleon gradient also generates a hyperfine-structure correction