Screened Scalar Fields in the Laboratory and the Solar System

The last few decades have provided abundant evidence for physics beyond the two standard models of particle physics and cosmology. As is now known, the by far largest part of our universe's matter/energy content lies in the `dark' and consists of dark energy and dark matter. Despite intensive efforts on the experimental as well as the theoretical side, the origins of both are still completely unknown. Screened scalar fields have been hypothesized as potential candidates for dark energy or dark matter. Among these, some of the most prominent models are the chameleon, symmetron, and environment-dependent dilaton. In this article, we present a summary containing the most recent experimental constraints on the parameters of these three models. For this, experimental results have been employed from the qBOUNCE collaboration, neutron interferometry, and Lunar Laser Ranging (LLR), among others. In addition, constraints are forecast for the Casimir And Non Newtonian force EXperiment (CANNEX). Combining these results with previous ones, this article collects the most up-to-date constraints on the three considered screened scalar field models.


I. INTRODUCTION
Dark energy (DE) and dark matter (DM) pose some of modern physics' greatest open problems.Modifications of general relativity (GR) in the form of scalar-tensor theories [1] in which a scalar field is coupled to the gravitational metric tensor in the matter action are frequently employed in attempts to solve these conundrums [2,3].While many of these theories lead to a universal coupling between the scalar field and the trace of the energy-momentum tensor of (Standard Model) matter and, consequently, a fifth fundamental force of Nature, such emergent effects are already tightly constrained within our Solar System [4][5][6].
Screening mechanisms offer phenomenologically appealing ways for circumventing these constraints by rendering scalar fifth forces feeble in matter at least as dense as our Solar System.
In this article, we present an overview of existing experimental and observational constraints on chameleon, symmetron and environment-dependent dilaton models.For this, we add the most recent constraints on chameleons [54,63] and symmetrons [56,59] to the summarised data from Ref. [31], and create plots combining constraints obtained from gravity resonance spectroscopy (qBounce), Lunar Laser Ranging (LLR) [58] and neutron interferometry [59] for the environmentdependent dilaton.In addition, we present projected constraints for all three models expected to be obtained from the upcoming Casimir And Non Newtonian force EXperiment (Cannex) [62,69].
In addition to providing a brief review of existing constraints, this article updates the qBounce constraints from Refs.[34,36,47,50] for the symmetron and chameleon models.Previous analyses assumed an ideal vacuum density of ρ V = 0, exclusively used perturbation theory to compute the energy shifts that a screened scalar field would induce on a neutron bouncing in a gravitational field (which is a good approximation for the chameleon field [70] but not for the symmetron field), and neglected parameter regions for which the symmetron field is in its symmetry-broken phase even inside the neutron mirror.This updated analysis employs the actual experimental residual gas density of ρ V = 2.32 × 10 −7 kg/m 3 in the vacuum chamber.For parameter regions where perturbation theory is inaccurate, a non-perturbative numerical treatment is used.Additionally, the symmetry-broken phase of the neutron is considered, as detailed in Ref. [51].Analogous to Ref. [58], assumptions necessary for the analysis as, e.g., neglecting higher order couplings between scalar field and matter as well as neglecting the influence of the vacuum chamber on the scalar field have been properly considered.In order to physically justify these assumptions, all derived constraints have been cut off wherever this is required.As demonstrated in Appendix A, each of these improvements corrects the previous symmetron constraints by several orders of magnitude.
The article is organized as follows.In Sec.II, we provide the required theoretical background on the considered screened scalar field models, while in Sec.III our way of obtaining the constraints is described.Next, in Secs.IV A-IV C, up-to-date constraints on the chameleon, the symmetron and the environment-dependent dilaton models are represented, respectively.Finally, in Sec.V, we draw our conclusions.

II. THEORETICAL BACKGROUND
In this article, we only discuss screened scalar field models with canonical kinetic terms in the Einstein frame in contrast to galileon models that require additional kinetic terms for implementing their screening via the Vainshtein mechanism [71].Consequently, all considered models for a scalar ϕ can be described by the following action where m pl is the reduced Planck mass, g µν the Einstein frame metric, gµν the Jordan frame metric, V (ϕ) denotes the scalar's self-interaction potential characterising each individual screened scalar field model (see Tab. I) and L SM the Lagrangian describing the Standard Model (SM) fields ψ i .
The conformal transformation between Jordan and Einstein frame is given by gµν = A 2 (ϕ)g µν , where the form of the conformal factor A(ϕ) is determined by the particular scalar field model, see Tab.I.In the Einstein frame, this leads to a universal coupling between scalar ϕ and the trace of the energy-momentum tensor T µν .In turn, this coupling leads to a fifth force of Nature, which, for a test particle with mass m in the non-relativistic limit is given by with the dimensionless coupling parameter The three considered scalar field models characterised by their particular forms of the potential V (ϕ) and the conformal factor A(ϕ). Chameleon: the different chameleon models are distinguished by the parameter n ∈ Z + ∪2Z − \{−2}.Λ has the dimension of a mass and parameterizes the self-interaction of the chameleon.The mass scale M c describes the chameleon's coupling to matter.Symmetron: µ is a tachyonic mass and λ S is a dimensionless self-coupling parameter.M is a mass scale describing the symmetron coupling to matter.Dilaton: The environment-dependent dilaton is characterised by a constant energy density V 0 , a dimensionless self-coupling parameter λ, and a dimensionless constant A 2 parameterizing the coupling to matter.
If the SM fields ψ i correspond to non-relativistic matter with density ρ, we can use T µ µ = ρ.In this case, the scalar is subject to an effective potential which can be derived from Eq. (1).Consequently, the equation of motion for the scalar ϕ is given by Depending on the different possible choices for V (ϕ) and A(ϕ) as given in Tab.I, the effective potential in Eq. ( 4) gives rise to the chameleon mechanism [11,12] or the Damour-Polyakov mechanism [15].While the former is characteristic for chameleon models, symmetrons are subject to the latter.Initially, the environment-dependent dilaton was also thought to be mainly screened by the Damour-Polyakov mechanism but in Ref. [58] it was demonstrated that for some parts of the dilaton parameter space the chameleon mechanism is actually dominant concerning the fifth force screening.

III. CONSTRAINT CALCULATION
In this section, further details concerning the newly computed or re-analyzed constraints presented in this article are provided.Wherever necessary, numerical algorithms have been employed to solve the scalar field equations of motion and, if applicable, the stationary Schrödinger equation.
Refs. [59,60] provide further details concerning these algorithms.Herein, new results or re-analyses are provided for the qBounce, neutron interferometry and Cannex laboratory experiments as well as Lunar Laser Ranging.In the subsequent subsections, further details concerning the constraints from each of these experiments are presented.In order to justify the neglect of any higher order couplings, the analysis herein is restricted to which requires excluding those regions of the constraint plots that do not fulfill this assumption.
For qBounce and Cannex, only interaction ranges in vacuum of up to 1 mm are considered, in which case any influence from the vacuum chamber can be neglected.Furthermore, for Cannex, parameters for which the field does not decay to its potential minimum inside the upper mirror are excluded, while for neutron interferometry an interaction range of maximally 0.25 mm within the cylindrical walls enclosing the chambers through which the superposed neutron beams traverse is considered.The latter is required to ensure that the minimum value ϕ M of the scalar field within the walls can be used as a boundary condition.

A. qBounce constraints
The qBounce experiment is a realization of gravity resonance spectroscopy [34,[72][73][74].It exploits the fact that ultracold neutrons, i.e. neutrons with very small kinetic energies, are totally reflected from most materials.In Earth's gravity, the quantum theoretical energy levels of a neutron are discrete and non-equidistant.This enables the realization of spectroscopy by inducing transitions between different energy levels.The very high precision enables effective searches for any hypothetical fifth forces.Specifically, measurements of the transition energies ∆E 13 := E 3 −E 1 and ∆E 14 := E 4 − E 1 have been carried out.The experimentally measured values are ∆E 13 = (1.9222± 0.0054) peV (expected value 1.9145 peV) and ∆E 14 = (2.6874± 0.0074) peV (expected value 2.676 peV), consistent with the transition energies expected from a Newtonian gravitational potential alone [47].If it exists, a hypothetical scalar field would induce an additional potential.
qBounce's experimental geometry, as realized in Ref. [47], can be approximated to an excellent degree by an infinitely extended single mirror, see e.g.Refs.[44,51], and the individual scalar field profiles computed by numerically solving the corresponding equations of motion (5).The transition energies are computed numerically by solving the stationary Schrödinger equation including Newtonian and scalar field effects where m n is the mass of the neutron, and the potential is given by with X ∈ {C, S, D} (as abbreviations for chameleon, symmetron and dilaton, respectively) and the screening charge Q X , which quantifies the amount of screening of the neutron, see Appendix C.
Furthermore, the same relations as in Ref. [59] have been employed herein.
Previous publications [47,50,58] explored two distinct approximations for the coupling of neutrons to screened scalar fields, both treating the neutron as a classical sphere.In the fermi screening approximation, the neutron's radius is set to R = 0.5 fm, a value grounded in QCD scattering data.Conversely, the micron screening approximation sets the neutron's radius at 5.9 µm derived from its wavefunction's vertical extent.In this investigation, we exclusively adhere to the fermi screening approximation.Notably, due to the absence of a well-defined micron screen- The current constraint criterion fails to account for correlations among ∆E 13 , ∆E 14 and the scalar field parameters under scrutiny.Due to persisting theoretical uncertainties in energy shift predictions, notably attributable to the screening charge approximation, conducting a rigorous statistical analysis is presently unfeasible.However, it has been carefully checked that this simplified constraint criterion is a very good approximation to a more rigorous χ 2 -analysis, see Appendix A 2, and is hence employed at the current level of theoretical description.

B. Neutron interferometry constraints
A neutron interferometer uses the wave nature of neutrons for performing interferometric measurements [74][75][76].Specifically, the path of a single propagating neutron is split into a superposition of two macroscopically separated beams before these paths are recombined and the neutron is finally detected.Each beam passes through a cylindrical chamber, one of which, denoted "air", is filled with air of ambient density to suppress the field, while the other, denoted "vacuum", actually contains helium with low pressure.In the latter, a scalar field typically takes on a larger field value than in the other chamber filled with air.If a scalar field indeed exists, the neutron would experience a different phase shift in each of the paths, which in the semi-classical limit is given by (see e.g.Ref. [59]) U X;P (r, z) dz , (10) with X ∈ {C, S, D}, k 0 being the wave number of the neutron, r the radius from the center at which the neutron beam propagates through the chamber, L the length of the chamber, U X;P (r, z) = Q X m n A X (ϕ) − A X (ϕ Air ) the scalar field potential in the chamber with pressure P , and the z-integration extending over the classical flight path (CFP) inside the corresponding chamber.
Experimentally, the following two measurement modes have been used to constrain scalar field parameters: 1. Profile mode In this mode, the following phase shift is evaluated ∆φ X;P = δφ X;P (0) − δφ X;P (0.015 m) , for both chambers, with the vacuum chamber having its pressure at 10 −4 mbar, which corresponds to the lowest pressure measured in the experiment.The experiment actually measured This quantity is negative since the potential is more suppressed close to the chamber walls and due to |∆φ X;Air | < |∆φ X;Vacuum |.The experiment constrains as will be further detailed in Appendix B 1.

Pressure mode
In this mode, the following quantity is measured instead with a vacuum pressure of P 0 = 2 × 10 −4 mbar and a reference pressure of P 1 = 10 −2 mbar.
Scalar field parameters are constrained if as is detailed in Appendix B 2.
Further details on the specific experimental setup used to obtain the constraints in the present article can be found in Ref. [59].

C. Computing observables for Lunar Laser Ranging
Lunar Laser Ranging (LLR) allows for the measurement of the Moon's orbit with high precision and consequently tests general relativity as well as potential deviations from it, for example, due to the presence of a scalar fifth force [77].For this, the Earth-Moon distance is measured by firing a laser beam at the retroreflectors that were installed on the Moon's surface during the Apollo missions and detecting the reflection back on Earth.The propagation time of the laser pulse during its travel from Earth to Moon and back provides the corresponding distance to high precision.For previous investigations employing LLR in the context of symmetron and chameleon fields, see Refs.[48,78,79].
The measured value for violations of the equivalence principle is [80] where a ϕ ♁ and a ϕ refer to the scalar field induced accelerations of the Earth and Moon towards the Sun, whereas a G is the regular Newtonian acceleration towards the Sun.For the constraint plots in this article, parameters are constrained for which the scalar field contribution leads to at least a 2σ-deviation of the measured value corresponding to The perihelion precession of the Moon has been computed, e.g. in Ref. [81], and leads to parameter constraints given by [82] δΩ where R is the maximum Earth-Moon separation and use has been made of D. Computing the pressure in the Cannex experiment Among the high-precision experiments is the Casimir And Non-Newtonian force EXperiment (Cannex), which has successfully completed its proof-of-principle phase and will soon commence operation [62,69].It is the first experiment designed to measure the Casimir force employing two plane parallel plates.Due to this special geometry, interfacial as well as gravity-like forces are maximized leading to increased sensitivity.This experiment allows to probe a wide range of dark sector forces, Casimir forces in and out of thermal equilibrium as well as gravity.
Cannex measures the pressure where ρ M and ρ V are the densities of the plates and of the surrounding vacuum, d is the distance between both plates and ϕ 0 (d) is the value of the scalar field in the middle between both plates, as well as the pressure gradient The latter approximation holds for small enough δ.The sensitivity of these measurements at one standard deviation (1σ) have already been analyzed in detail as function of d in Ref. [69].Herein, parameter constraints are derived for For the derivation of prospective constraints, it has been used that the plate separation and vacuum density can be varied between 3 µm < d < 30 µm and 5.3 × 10 −12 kg/m 3 < ρ V < 2.6 kg/m 3 , respectively.More details about this experiment can be found in Ref. [58].

IV. RESULTS
This section provides succinct discussions of the obtained constraints for the chameleon, symmetron and dilaton model.

A. Chameleon constraints
In Fig. 1, all currently available chameleon constraints on the parameters n and M c for Λ being set to the DE scale of 2.4 meV are shown as well as those on the parameters Λ and M c for the case For this, the analytically exact solutions from Ref. [41] are used to compute the prospective Cannex constraints, while the one-mirror solution for qBounce is computed numerically.
Our analysis shows that qBounce has set 'weak' constraints for n = 1 but no constraints for Λ = 2.4 meV and small positive n.The large discrepancy to previous chameleon constraints published in Refs.[34,36] are primarily rooted in a different definition of the screening charge, further elaborated on in Appendix C.
For Cannex, our analysis predicts constraints both for n = 1 as well as for other small values of n if Λ = 2.4 meV.However, constraints on parts of the parameter space that were not previously constrained by other experiments can only be obtained in the latter case.Neutron interferometry is currently not capable of producing new constraints for the cases considered in Fig. 1.A static chameleon field always fulfills the inequality FIG. 1: Constraints on the parameter space of chameleon models based on the review [31] and including results from quantum Casimir pressure [63], levitated force sensor measurements [54], and, for qBounce and Cannex (the last only prospective), the analysis made in this article.a) Here, the parameter Λ has been fixed to the DE scale of 2.4 meV.The blue area shows the combined prospective constraints of pressure and pressure gradient measurements on chameleon interactions resulting from Cannex.qBounce and neutron interferometry can set no constraints.This figure was adapted from Ref. [62].b) The blue area shows the prospective Cannex constraints for the chameleon model with n = 1, which are expected to fully overlap with already existing constraints.
Constraints from qBounce are depicted in the small dark red area that overlaps with atomic measurements and Cannex.The strong discrepancy of the computed qBounce constraints in this work and previous constraints obtained in Refs.[34,36] is further elucidated in Appendix C.
The quantity where L = 9.4 cm, can be computed analytically and used to bound the largest expected phase shift.It should be noted that as well as Over all unconstrained parts of the parameter space one has Consequently, no new constraints can be obtained from neutron interferometry.Obtained constraints would be many orders of magnitude 'weaker' than existing constraints, rendering a more detailed analysis for this experiment futile in the case of chameleons.

B. Symmetron constraints
All existing constraints on the symmetron model derived from tabletop experiments are depicted in Fig. 2. Notably, astrophysical experiments are unable to impose constraints on the µ values discussed in this article.
For the re-analysis of the qBounce constraints originally presented in Ref. [47,50], the analytically exact solutions of Refs.[44,51] are employed.However, for the prospective Cannex constraints, the two mirror solution is computed by solving the differential equation numerically.
The constraints are based on the most recent analysis of Cannex in Ref. [62].Constraints from neutron interferometry are based on Ref. [59].In Appendix A 1, we delineate how the newly computed symmetron constraints compare to those previously derived in Ref. [47], elucidating the impact of each innovation on the resulting constraints.
FIG. 2: Constraints on the parameter space of the symmetron model for different values of the tachyonic mass µ are depicted based on the review [31], containing Eöt-Wash results [83] and the atom interferometry analysis from Ref. [43]; investigations of hydrogen, muonium and the electron (g-2) [56]; and our new analysis for qBounce, neutron interferometry and Cannex (the last only prospective).a) Most table-top experiments are sensitive to the µ values displayed here since the symmetron range is close to 1 mm.b) Cannex is expected to lead to dominant constraints since the range is approximately 1 µm, which is close to the plate separation.Eöt-Wash and atom interferometry are currently not able to set constraints for these values of µ. c) Only some quantum experiments can still set constraints for these interaction ranges.d) For these values of µ, the screening of nucleons is strong enough that they reach their limit for probing symmetrons, which affects the qBounce and hydrogen constraints.In contrast, Ref. [56] argues that muonium is always unscreened, due to its composition of effectively point-like particles, and can hence still set substantial constraints.

C. Dilaton constraints
All existing constraints on the environment-dependent dilaton model were originally obtained in Refs. [58,59] and are based on experimental results from qBounce, LLR, and neutron interferometry.Prospective constraints for Cannex were also presented in Ref. [58].A combined summary of these constraints can be found in Fig. 3.

V. CONCLUSIONS
In this article, the most recent experimental constraints on the parameter spaces of some of the most popular screened scalar field models are summarized, i.e., the chameleon, symmetron and environment-dependent dilaton.Among others, experimental results from the qBounce collaboration, neutron interferometry, and LLR have been employed.In addition, future constraints for Cannex are predicted.The results herein collect the most up-to-date constraints on the considered screened scalar field models.Furthermore, several improvements to previously obtained constraints have been made.In some cases, this led to deviations by several orders of magnitude compared to previously obtained results.
turbed wave function to perceive an almost constant potential shift, aside from the region 0.1 µm immediately above the mirror, which contributes only minimally to the integral.This results approximately in where U V EV X is the value of U X (z) corresponding to the VEV.However, strong scalar fields at such short ranges distort the neutron wave functions, resulting in significantly different energy shifts for various energy states, which would be readily observable in experiments.The numerical solution of the Schrödinger equation provided herein accounts for that and hence allows to extend previous constraints.
Thus, each improvement in the present analysis significantly changes existing constraints.In contrast, a full statistical data analysis to obtain constraints can safely be neglected at the current level of theoretical description, as will be demonstrated in the following section.
Ref. [50] Line A Line B FIG. 4: Comparison between qBounce constraints obtained in Ref. [50] and the constraints computed herein.a) The symmetron parameter µ has been set to 0.1 eV.Left of line A, no symmetron solutions exist, since the symmetron is in its symmetric phase inside the vacuum as well as mirror regions.Right of line B, the symmetron is in its symmetry-broken phase inside the mirror.b) Here, µ has been set to 1 keV, a value for which no constraints have been obtained in Ref. [50].

Constraint criteria for qBounce
In Ref. [47], constraints on the symmetron model from qBounce have been established through a comprehensive χ 2 -data analysis, considering correlations between ∆E 13 , ∆E 14 as well as each symmetron parameter µ, λ S and M .However, this analysis is based on different theoretical assumptions, including a vacuum density of ρ V = 0, exclusive utilization of perturbation theory for computing energy shifts, neglecting the symmetry-broken phase of the symmetron inside the neutron mirror and employing different cutoff criteria.Here, we replicated this earlier investigation under the previously made assumptions but employed the constraint criterion outlined in Eq. ( 9) instead of a full χ 2 -data analysis.This allows to assess the error incurred by forgoing a comprehensive statistical examination.As demonstrated in Fig. 5, this error is negligible in a log-log plot, validating our adoption of a simpler constraint criterion.The shaded regions illustrate parameter constraints (employing the fermi screening approximation) derived in Ref. [47] for various fixed values of M .Dotted lines mark the boundary of the constraint regions computed with the simplified constraint criterion outlined in Eq. ( 9).The constraints depicted in this plot were computed using previously made theoretical assumptions, as is detailed in Appendices A 1 and A 2. Updated constraints using the most recent theoretical assumptions are depicted in Fig. 2.
where µ M represents the chameleon or dilaton mass, respectively, within a neutron, µ V denotes the corresponding mass within the surrounding vacuum and R is the radius of the sphere.This definition ensures that 0 , for "screened" bodies with µ M R → ∞ , 1 , for "unscreened" bodies with µ M R → 0 . (C2) In contrast, Ref. [36] calculated qBounce chameleon constraints using an alternative definition of the screening charge, sometimes treating the neutron as a test particle even when µ M R ≫ 1, leading to significantly larger constraints.Ref. [34], on the other hand, has neglected neutron screening altogether.This disparity underscores the imperative for further theoretical advancements to transcend heuristic approximations and accurately determine the true coupling of neutrons to the individual scalar field.The analysis of the symmetron field herein uses the screening charge derived in Refs.[44,51], which has already been employed in the original analyses in Refs.[47,50].For some further background information concerning the methodology of screening charges, see e.g.

FIG. 3 :
FIG.3: Re-computed combined constraints from Refs.[58,59] on the parameters of the environment-dependent dilaton are presented.The interaction range of the dilaton with log 10 (V 0 /MeV 4 ) = 1 is plotted for two different vacuum densities, the 1 µm and 1 AU contours correspond to ρ V = 2.32 × 10 −7 kg/m 3 (qBounce) and ρ V = 1.67 × 10 −20 kg/m 3 (interplanetary medium), respectively.a) The constraint areas shift towards lower values of λ as V 0 increases without altering their shape.This phenomenon occurs because the fifth force effectively relies on the product of V 0 and λ rather than their individual values in this regime.b) The constraint areas exhibit systematic shifts and cuts as illustrated by the indicated arrows.This figure has been adapted from Ref.[58].
FIG.5:The shaded regions illustrate parameter constraints (employing the fermi screening ap-