Antimatter Free-Fall Experiments and Charge Asymmetry

Abstract

We propose a method by which one could use modified antimatter gravity experiments in order to perform a high-precision test of antimatter charge neutrality. The proposal is based on the application of a strong, external, vertically oriented electric field during an antimatter free-fall gravity experiment in the gravitational field of the Earth. The proposed experimental setup has the potential to drastically improve the limits on the charge-asymmetry parameter ${\overline{ϵ}}_q$ of antimatter. On the theoretical side, we analyze possibilities to describe a putative charge-asymmetry of matter and antimatter, proportional to the parameters ${ϵ}_q$ and ${\overline{ϵ}}_q$, by Lagrangian methods. We found that such an asymmetry could be described by four-dimensional Lorentz-invariant operators that break CPT without destroying the locality of the field theory. The mechanism involves an interaction Lagrangian with field operators decomposed into particle or antiparticle field contributions. Our Lagrangian is otherwise Lorentz, as well as PT invariant. Constraints to be derived on the parameter ${\overline{ϵ}}_q$ do not depend on the assumed theoretical model.

1. Introduction

The CPT theorem [1,2] encompasses large classes of local field theories with Lorentz-invariant, local, and Hermitian Lagrangian terms. All theories investigated in [1,2] are invariant under the combined action of charge conjugation (C), parity inversion (P), and time reversal (T). Due to the (almost) universal character of the theories studied in [1,2], the accepted assumption is that CPT invariance in an inalienable symmetry of nature. Here, we explore the possibility that, by enlarging the class of the possible interactions, it is possible to evade the CPT theorem by adding Hermitian, local, yet CPT-violating interactions that lead to long-range, gravity-mimicking, CPT-violating interactions, and still conserve Lorentz invariance.

Such interactions have been investigated in [3,4,5,6] with a separate charge asymmetry parameter for antimatter being introduced on an ad hoc basis. We, here, explore if such asymmetry parameters could be introduced via a separation of the quantum-field theoretical fermion operators into positive-energy and negative-energy components, and we discuss a possible related experimental ansatz.

Such interactions violate a fundamental principle of the construction of quantum field theories, namely, gauge invariance. This is the primary reason why one can introduce putative charge-asymmetry of matter and antimatter by four-dimensional Lorentz-invariant operators that break CPT without destroying the locality of the field theory. Our interaction terms describe the exchange of virtual bosons but exclude virtual annihilation channels. While, with this approach, several difficulties associated with nonlocal and Lorentz-violating field theories can be avoided, gauge invariance is violated.

The experimental proposal discussed here is independent of the theoretical model introduced; the aim is to set limits for the charge asymmetry parameter for antimatter, which could otherwise be introduced on an ad hoc basis [3,5,6]. Our approach is based on the explicit assumption that matter–antimatter symmetry perfectly holds for the gravitational interaction. There are strong theoretical arguments to support this assumption [7,8]. This conjecture also is compatible with the experimental results reported in [9], which will hopefully be improved in the near future.

From a phenomenological point of view, the decisive characteristic of our models is the fact that they allow for the presence of a different charge asymmetry parameter for matter versus antimatter. Such effects have been discussed in the literature, independent of a quantum-field theoretical underlying formulation [4,5,6]. Our proposal for an improvement of the charge asymmetry parameter for antimatter is connected with antimatter gravity experiments with the aim of comparing a residual electrostatic force on an antihydrogen atom with the gravitational force.

Tests of the charge neutrality of matter (and antimatter) have recently attracted considerable attention [4,5,6]. A conceivable electric charge of the neutron was investigated in [10]. An excellent review on various theoretical models allowing for charge asymmetry is presented in [11]. An essential assumption underlying our investigations is that the gravitational interaction for antimatter in the gravitational field of the Earth is the same as for matter. There are strong theoretical arguments to support this initial assumption [7,8,12].

Past efforts [13,14,15] to measure the gravitational interaction of electrons and positrons have suffered from the problem of eliminating electric-field effects. This has been a tremendous problem that has never been solved convincingly in experiments [14,15]. It is connected with the elimination of the influence of the electric field generated by the electrons in the metal or other material (drift tube) surrounding the fall line of the charged leptons.

Theoretical arguments [13] suggested that, under the presence of the electric field generated by the electrons, the total gravitational acceleration of electrons converges to zero, while that of positrons would be twice the acceleration due to gravity. Indeed, experiments with positrons were not succesfully reported despite considerable invested effort [13,14,15]. We argue the other way and show that, in a gravitational experiment carried out with (supposedly) electrically neutral particles, the influence of any deliberately introduced, strong, external, electric field allows one to display the effect of a residual charge excess, leading to a test of charge neutrality.

The influence of a hypothetical charge excess in antimatter on the dynamics can be explored effectively in a strong, uniform, vertical external electric field. This is because any potential charge asymmetry is compared to the extremely weak gravitational force. Our estimates, reported in this article, suggest that limits on the charge asymmetry of antimatter could be improved by many orders of magnitude if an antihydrogen gravity experiment is done in the presence of a strong external electric field. SI mksA units are used in this article unless stated otherwise.

2. Charge Symmetry and Gravity

Let us assume that electron and proton charges do not quite add up to zero so that there is an ever so slight residual charge to be associated with a hydrogen atom, an idea originally formulated by Einstein at the 1924 Lucerne Meeting of the Swiss Physical Society as was explicitly mentioned in [16,17]. In accordance with [4], we parameterize a putative charge excess as follows,

$$\begin{array}{cc}\hfill q_e=& -|e|,q_p=|e|(1+{ϵ}_{p-e}),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {ϵ}_{p-e}=& \frac{q_e+q_p}{|e|},{ϵ}_n=\frac{q_n}{|e|}.\hfill \end{array}$$

(2)

Here, $q_e=e$ is the electron charge, and $|e|$ is its modulus, while $q_p$ and $q_n$ are the proton and neutron charges, respectively. If we assume, with [4], charge conservation in the $\beta$ decay of the neutron, then the charge-neutrality violating parameter ${ϵ}_n$ for the neutron (let us be clear that the neutron acquires an infinitesimal electric charge under the assumptions made in [4]) becomes

$${ϵ}_n={ϵ}_{p-e}\equiv {ϵ}_q.$$

(3)

If a body containing Z protons and electrons, as well as N neutrons is measured as being neutral with sensitivity $\delta q$, the one can obtain a limit on $|{ϵ}_q|$, which is on the order of

$$\begin{array}{cc}\hfill \left|Z{ϵ}_{p-e}+N{ϵ}_n\right|=& (Z+N)|{ϵ}_q|\le \delta q,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill |{ϵ}_q|\le & \frac{\delta q}{(Z+N)|e|}.\hfill \end{array}$$

(4b)

The essential idea of the acoustic method used in [4] is that, under the assumption of a nonvanishing charge asymmetry in matter, electromagnetic waves incident on an electrically neutral gas would set the gas atoms in motion, inducing sound waves. However, this method of determining limits on $|{ϵ}_q|$ is not free from pitfalls and requires a considerable additional mathematical formalism in the evaluation of the experiment. For example, according to a note to Table I of [18], data published in the previous work [19] may exhibit inconsistencies.

A paper [20] that initially claimed an accuracy on the level of ${10}^{-23}$ for $|{ϵ}_q|$ has recently been questioned in [4], with the claim that their result on $|{ϵ}_q|$ could not be considered to be better than ${10}^{-19}$ if all inaccuracies and neglected systematic effects in the paper [20] are properly taken into account. The paper [4] also indicates additional rectification of the analysis of the resonant modes in the gas-filled capacitor used in the previous experiment [20]. Table I of [4] contains a comprehensive compilation of previous measurements of ${ϵ}_q$. We will use their result, given in an unnumbered equation on the second-to-last page of [4],

$${ϵ}_q=(-0.1\pm 1.1)\times {10}^{-21},$$

(5)

for matter particles (both a hydrogen atom as well as the constituent atoms of the Earth). Limits on the charge asymmetry of matter have also been derived on the basis of model-dependent astrophysical methods [21,22]. Separate investigations put limits on the neutrality of neutrinos on the basis of astrophysical observations [23,24,25,26,27].

In contrast, the constraints on charge neutrality for antimatter are looser by many orders of magnitude. Tests on the electric charges of positrons and antiprotons can be derived from measurements of their cyclotron resonance frequencies and from spectroscopic data [3]. The most recent direct tests [5,6] revealed a $1\sigma$ limit $68.3$% confidence level)

$$|{\overline{ϵ}}_q|\le 7.1\times {10}^{-10}$$

(6)

for antimatter. Here, the parameters for antiparticles are given by

$$\begin{array}{cc}\hfill q_{\overline{e}}=& |e|,q_{\overline{p}}=-|e|(1+{ϵ}_{\overline{p}-\overline{e}}),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {ϵ}_{\overline{p}-\overline{e}}=& \frac{q_{\overline{e}}+q_{\overline{p}}}{-|e|},{ϵ}_{\overline{n}}=\frac{q_{\overline{n}}}{-|e|}.\hfill \end{array}$$

(8)

The parameters $\overline{e}$, $\overline{p}$, and $\overline{n}$ stand for the positron, antiproton, and antineutron, respectively. We shall also make the assumption that charge is conserved in the $\beta$ decay of the antineutron and write

$${ϵ}_{\overline{n}}={ϵ}_{\overline{p}-\overline{e}}\equiv {\overline{ϵ}}_q.$$

(9)

Furthermore, we shall assume, as demonstrated in a number of experiments [28,29,30], that gravitational and gravity-like interactions are equivalently realized on the microscopic (atomic) and macroscopic level.

The electric charge asymmetry of antihydrogen, if it exists, is not necessarily opposite to that found in hydrogen. However, there might be good arguments to support this conjecture. Namely, electrons and positrons constitute a particle–antiparticle pair and, therefore, are described by the same Dirac equation, which predicts that the electric charges of the positron and electron add up to zero. The same applies to protons and antiprotons. However, one observes that leptons and hadrons are still two completely different particle species. Therefore, it could appear easier to speculate about the broken electric neutrality of a hydrogen atom rather than the broken electric neutrality of, say, positronium.

3. Charge Asymmetry and CPT Violation

Typically (see [4,16,17]), the charge-asymmetry parameters ${ϵ}_q$ and ${\overline{ϵ}}_q$ are used as ad hoc parameters without any attempt being made to formulate an underlying quantum field theory that could describe the charge-symmetry violation. Let us attempt to realize somewhat higher ambitions and explore candidate models. In the field-theoretical sense, let us explore if a slight breaking of the charge symmetry could be formulated in terms of a gauge-symmetry breaking (GB) modification of the quantum electrodynamic (QED) interaction Lagrangian. Denoting field operators by a hat, we write the Lagrangian (we temporarily switch to the natural unit system with $\hslash =c={ϵ}_0=1$, as is customary in particle physics)

$$\begin{array}{cc}\hfill {\widehat{\mathcal{L}}}_{\mathrm{GB}}=& \sum _{f=e,p,n}{\widehat{\overline{\psi }}}_f(\mathrm{i}{\gamma }^{\mu }{\partial }_{\mu }-m){\widehat{\psi }}_f-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }\hfill \\ \hfill & -e\left({\widehat{\overline{\psi }}}_e{\gamma }^{\mu }{\widehat{\psi }}_e-{\widehat{\overline{\psi }}}_p{\gamma }^{\mu }{\widehat{\psi }}_p\right){\widehat{A}}_{\mu }\hfill \\ \hfill & -{ϵ}_q|e|\left({\widehat{\overline{\psi }}}_p{\gamma }^{\mu }{\widehat{\psi }}_p+{\widehat{\overline{\psi }}}_n{\gamma }^{\mu }{\widehat{\psi }}_n\right){\widehat{A}}_{\mu }.\hfill \end{array}$$

(10)

Here, the electron-positron field operator is ${\psi }_e$, while the composite spin-1/2 operator for the proton–antiproton field is ${\psi }_p$, and the neutron–antineutron field could be described by a spin-1 generalization of the Dirac equation (see [31]). The quantized photon field is described by the four-vector field operator ${\widehat{A}}_{\mu }$, which enters the field-strength tensor operator ${\widehat{F}}_{\mu \nu }={\partial }_{\mu }{\widehat{A}}_{\nu }-{\partial }_{\nu }{\widehat{A}}_{\mu }$, where ${\partial }_{\mu }\equiv \partial /\partial x^{\mu }$ is the partial derivative with respect to the space-time coordinate $x^{\mu }$. In the form of ${\widehat{\mathcal{L}}}_{\mathrm{GB}}$ given in Equation (10), we use the assumption (9). As a side remark, we note that the current conservation implies that ${ϵ}_{p-e}=2{ϵ}_u+{ϵ}_d$, where the valence quark couplings for the up and the down quark are ${ϵ}_u$ and ${ϵ}_d$, respectively.

From the form of the Lagrangian (10), it follows that electrons (and consequently positrons) carry a charge $\pm e$, while protons (and antiprotons) carry a charge $\pm (1+{ϵ}_{p-e})|e|$. This results in a hydrogen atom having a charge ${ϵ}_{p-e}|e|$, while antihydrogen atoms carry a charge $(-{ϵ}_{p-e}|e|)$. One might, therefore, assume that ${ϵ}_q=-{\overline{ϵ}}_q$.

As there exist very stringent limits on ${ϵ}_q$ (for particles, see [4]), it is interesting to explore the possibility of different charge-asymmetry parameters for particles and antiparticles, i.e., a Lagrangian with two different parameters ${ϵ}_q$ and ${\overline{ϵ}}_q$. One may, thus, explore the phenomenological consequences of the Lagrangian

$$\begin{array}{cc}\hfill {\widehat{\mathcal{L}}}_{\mathrm{SYM}}=& \sum _{f=e,p,n}{\widehat{\overline{\psi }}}_f\left(\mathrm{i}{\gamma }_f^{\mu }{\partial }_{\mu }-m\right){\widehat{\psi }}_f-\frac{1}{4}{\widehat{F}}_{\mu \nu }{\widehat{F}}^{\mu \nu }\hfill \\ \hfill & -e\left({\widehat{\overline{\psi }}}_e{\gamma }_e^{\mu }{\widehat{\psi }}_e-{\widehat{\overline{\psi }}}_p{\gamma }_e^{\mu }{\widehat{\psi }}_p\right){\widehat{A}}_{\mu }\hfill \\ \hfill & -{ϵ}_q|e|\left({\widehat{\overline{\psi }}}_p^{(+)}{\gamma }_p^{\mu }{\widehat{\psi }}_p^{(+)}+{\widehat{\overline{\psi }}}_n^{(+)}{\gamma }_p^{\mu }{\widehat{\psi }}_n^{(+)}\right){\widehat{A}}_{\mu }\hfill \\ \hfill & +{\overline{ϵ}}_q|e|\left({\widehat{\overline{\psi }}}_p^{(-)}{\gamma }_n^{\mu }{\widehat{\psi }}_p^{(-)}+{\widehat{\overline{\psi }}}_n^{(-)}{\gamma }_n^{\mu }{\widehat{\psi }}_n^{(-)}\right){\widehat{A}}_{\mu }.\hfill \end{array}$$

(11)

Here, we have induced CPT violation by decomposing a general fermionic field operator $\widehat{\psi }\left(x\right)$ into a positive-frequency matter contribution ${\psi }^{(+)}\left(x\right)$ and a negative-frequency antimatter contribution ${\psi }^{(-)}\left(x\right)$, as follows,

$$\begin{array}{cc}\hfill \widehat{\psi }\left(x\right)=& {\psi }^{(-)}\left(x\right)+{\psi }^{(+)}\left(x\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\widehat{\psi }}^{(+)}\left(x\right)=& \sum _s\int \frac{{\mathrm{d}}^{3}p}{{\left(2\pi \right)}^3}\frac{m}{E}{a}_s\left(\overrightarrow{p}\right)u_s\left(\overrightarrow{p}\right){\mathrm{e}}^{-\mathrm{i}p·x},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\widehat{\psi }}^{(-)}\left(x\right)=& \sum _s\int \frac{{\mathrm{d}}^{3}p}{{\left(2\pi \right)}^3}\frac{m}{E}{b}_s^{+}\left(\overrightarrow{p}\right)v_s\left(\overrightarrow{p}\right){\mathrm{e}}^{\mathrm{i}p·x}.\hfill \end{array}$$

(14)

In view of the impossibility to transform positive-energy states into negative-energy states via Lorentz transformations (due the mass gap for Dirac particles), this decomposition is Lorentz invariant (for details, see [32]). It is useful to remark that the decomposition is also PT invariant. The decomposition of the effective proton and neutron field operators into positive-frequency and negative-frequency contributions has led to CPT violation and allows for different charge-asymmetry parameters ${ϵ}_q$ (for particles, i.e., for hydrogen) and ${\overline{ϵ}}_q$ (for antiparticles, i.e., for antihydrogen).

A closer inspection reveals that the CPT-violating terms in Equation (11) (those proportional to ${ϵ}_q$ and ${\overline{ϵ}}_q$) only modify the exchange of virtual photons between protons (and neutrons, due to the infinitesimal neutron charge); however, they do not lead to any annihilation of a proton–antiproton pair into a virtual photon. In that sense, the physics deduced from the interaction (11) implies a slight modification of the electromagnetic interaction between protons as compared to electrons, affecting the charge neutrality of the hydrogen and antihydrogen atoms.

This leads to a violation of the electromagnetic gauge invariance (both because of the slightly different electron and proton charge and because of the separation of the field operator in the interaction Hamiltonian into positive-energy and negative-energy components). The consequences of this assumption of astrophysical scales need to be explored. Two aspects can be mentioned here.

First, let us remember that a conceivable charge asymmetry of matter has been discussed as a possible (partial) explanation for the expansion of the Universe [33,34]. Even if the commonly accepted explanation involves a nonvanishing cosmological constant [35,36], it would be interesting to explore conceivable additional contributions to the expansion of the Universe due to antimatter and matter charge asymmetries.

There is a second aspect that might be even more interesting. Namely, a closer inspection reveals that the infinitesimal charge excess of the proton against the electron mimics, in the nonrelativistic limit, a gravitational interaction (see also Equation (16) below). However, the relativistic corrections to the gravitational interaction are known to be different for gravity as compared to electromagnetism [12,37].

If a nonvanishing charge asymmetry parameter were found for antimatter or matter, then one might need to investigate the effective modifications of the gravitational interaction (now adjusted for the retardation corrections to the electromagnetic admixtures due to the charge asymmetry). These could potentially have interesting connections to the observed discrepancies between astrophysical observations and the accepted theory of gravitation and, thus, to the dark matter problem.

The free photon field term in the Lagrangian density (11) involves the field-strength tensor

$${\widehat{F}}_{\mu \nu }={\partial }_{\mu }{\widehat{A}}_{\nu }-{\partial }_{\nu }{\widehat{A}}_{\mu }.$$

(15)

A closer inspection reveals that, in the nonrelativistic limit, the Lagrangian (11) leads to an effective interaction between hydrogen (H) and antihydrogen atoms ($\overline{\mathrm{H}}$) of (we now switch back to SI mksA units for the remainder of this article)

$$\begin{array}{cc}\hfill V_{\mathrm{H}\mathrm{H}}\left(R\right)=& \frac{{ϵ}_q^2{e}^2}{4\pi {ϵ}_{0}R},\hfill \end{array}$$

(16a)

$$\begin{array}{cc}\hfill V_{\overline{\mathrm{H}}\mathrm{H}}\left(R\right)=& \frac{{ϵ}_q{\overline{ϵ}}_q{e}^2}{4\pi {ϵ}_{0}R},\hfill \end{array}$$

(16b)

$$\begin{array}{cc}\hfill V_{\overline{\mathrm{H}}\overline{\mathrm{H}}}\left(R\right)=& \frac{{\overline{ϵ}}_q^2{e}^2}{4\pi {ϵ}_{0}R},\hfill \end{array}$$

(16c)

where R is the interatomic distance. None of the interaction Lagrangians investigated in [1,2] involve the splitting of a field operator into positive-frequency and negative-frequency parts. Our explicit separation of the field operators into positive- and negative-energy components in Equation (11) offers the possibility of introducing different charge asymmetry parameters for matter and antimatter.

4. Possible Implications on Charge Asymmetry of Antimatter

Based on the Lagrangian (11), our goal was to investigate whether or not the limit given in Equation (6), namely, $|{\overline{ϵ}}_q|\le 7.1\times {10}^{-10}$ could be improved by a simple experimental arrangement: Our suggestion was to place freely falling antihydrogen atoms into a uniform, vertically oriented, electric field, which would enable us to compare the gravitational force acting on the (supposedly) neutral antihydrogen atom to the gravitational force. We would, thus, intend to make the background electric field large, not small (as was otherwise attempted in [13,14,15]).

An estimate of the achievable accuracy of $|{\overline{ϵ}}_q|$ can be obtained by a simple calculation (see also Figure 1). The magnitude of the gravitational force on an antihydrogen atom is

$$|{\overrightarrow{F}}_g|=m_{\overline{\mathrm{H}}}g,$$

(17)

where $m_{\overline{H}}$ is the antihydrogen atom’s mass. The magnitude of the residual electric force on the antihydrogen atom is

$$|{\overrightarrow{F}}_e|={\overline{ϵ}}_q|e||\overrightarrow{E}|,$$

(18)

where $|\overrightarrow{E}|$ is the (possibly strong) external, vertically oriented electric field. Let us assume that we can experimentally establish that

$$|{\overrightarrow{F}}_e|<\chi |{\overrightarrow{F}}_g|,$$

(19)

i.e., that the residual putative electric force is less than a fraction $\chi$ of the gravitational force. Let us also parameterize the field strength of the vertically oriented field as $|\overrightarrow{E}|=|\overrightarrow{E}{|}_{\mathrm{SI}}\frac{\mathrm{V}}{\mathrm{m}}$, where $|\overrightarrow{E}{|}_{\mathrm{SI}}$ is the magnitude of the field, measured in volts per meter. This sets a limit on $|{\overline{ϵ}}_q|$, which is of the order of

$${\overline{ϵ}}_q<\frac{\chi m_{\overline{\mathrm{H}}}g}{|e||\overrightarrow{E}|}=1.02\times {10}^{-7}\frac{\chi }{|\overrightarrow{E}{|}_{\mathrm{SI}}}.$$

(20)

Typical Cockroft–Walton voltage multipliers operate at voltages of of $2\times {10}^4\mathrm{V}$ or more [38,39,40], and recent developments easily reach the range or ${10}^5\mathrm{V}$ (see [41]). Powerful tandem accelerators are known to operate at $2.5\times {10}^7\mathrm{V}$ (see [42,43]). In view of Paschen’s law [44], the electric breakdown strength inside an antihydrogen trap is of no concern in a trap held at a good vacuum (below ${10}^{-7}$ atmospheric pressure). Here, we estimate, somewhat conservatively, that an electric field strength of $|\overrightarrow{E}|={10}^6\frac{\mathrm{V}}{\mathrm{m}}$ can be realized in a dedicated experiment, corresponding to a value of $|\overrightarrow{E}{|}_{\mathrm{SI}}={10}^6$.

Under the further conservative assumption that the experiment establishes that the magnitude residual electric force is less than 10% of the gravitational force ($\chi =0.1$), one could improve the limit on ${\overline{ϵ}}_q$ into the range

$$|{\overline{ϵ}}_q|\lesssim {10}^{-14},$$

(21)

potentially improving the limit (6) by many orders of magnitude, leading to a drastic improvement of the charge neutrality parameter for antimatter. Possible limitations due to systematic effects encountered in the experiment (interatomic interactions and the spin coupling to the electric field) are discussed in the Appendix A.

5. Conclusions

In this work, we discussed possible antimatter charge asymmetry characterized by an antimatter charge asymmetry parameter ${\overline{ϵ}}_q$. Our considerations were motivated by the fact that the charge-asymmetry parameter ${\overline{ϵ}}_q$ for antimatter might be different from the charge asymmetry parameter ${ϵ}_q$ for particles (see Equation (11)). We, thus, advocate an electrically counterbalanced antimatter gravity for the potential determination of an improved bound on the charge-asymmetry parameter ${\overline{ϵ}}_q$ for antimatter.

In the proposed experimental setup, a strong, vertically oriented, electric field is placed in the line of free fall of an antihydrogen atom. Any putative charge asymmetry of antimatter, under the influence of the additional external field, would modify the resultant acceleration of the antihydrogen atom, which, in the absence of charge asymmetry, would exclusively be due to gravity. Even under pessimistic parametric estimates, the bounds on ${\overline{ϵ}}_q$ could potentially be improved by many orders of magnitude when compared to the current limits (see Equations (6) and (21) and [5,6]).