# Dilaton Solutions for Laboratory Constraints and Lunar Laser Ranging

Philippe Brax<EMAIL_ADDRESS>Institut de Physique Théorique,
Université Paris-Saclay, CEA, CNRS, F-91191 Gif/Yvette Cedex, France  Hauke
Fischer<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, A-1020 Wien, Austria  Christian Käding
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, A-1020 Wien, Austria  Mario Pitschmann
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, A-1020 Wien, Austria

###### Abstract

We derive approximate analytical solutions to the environment-dependent
dilaton field theory equations in the presence of a one or two mirror system
or a sphere. The one-dimensional equations of motion are integrated for each
system. The solutions obtained herein can be applied to qBOUNCE experiments,
neutron interferometry and for the calculation of the dilaton field induced
”Casimir force” in the Cannex experiment as well as for Lunar Laser Ranging.
They are typical of the Damour-Polyakov screening mechanism whereby deviations
from General Relativity are suppressed by a vanishingly small direct coupling
of the dilaton to matter in dense environments.

###### pacs:

98.80.-k, 04.80.Cc, 04.50.Kd

##  I Introduction

The accelerated expansion of the Universe may require the introduction of
additional degrees of freedom (see Joyce:2014kja  for a recent review). Such
new degrees of freedom, in particular light scalars, are theoretically well
motivated irrespective of their role for the acceleration of the expansion of
the Universe. If they exist in Nature, they must either only be feebly
coupling to other matter or appear in some screened form in order to prevent
detection in all past experiments and observations involving scalar fifth
forces. A number of screening mechanisms exist Joyce:2014kja , the chameleon
Khoury:2003rn ; Khoury:2003aq ; Brax:2004qh  and Damour-Polyakov mechanisms
Damour:1994zq , the K-mouflage Babichev:2009ee ; Brax:2012jr ; Brax:2014wla
and Vainshtein ones Vainshtein:1972sx , allowing such hypothetical fields to
remain unseen in local tests of gravity.

In the companion paper Pitschmann:2018aa , gravity resonance spectroscopy
Abele:2009dw ; Jenke:2011zz , a Casimir force experiment Sedmik:2018kqt  and
Lunar Laser Ranging (LLR) nordtvedt2001lunar  are used for the first time to
put new bounds on the environment-dependent dilaton Brax:2010gi ; Brax:2011ja
. The LLR bounds have already been discussed for chameleon PhysRevD.97.104044
and symmetron scalar fields PhysRevD.84.103521 . The dilaton model conjugates
two ingredients. First of all, it involves an exponentially decreasing
potential as expected in the strong coupling limit of string theory
Gasperini:2001pc ; Damour:2002nv ; Damour:2002mi  and second it is the
simplest realisation of the least coupling principle as advocated in
Damour:1994zq . Hence this model is a prime example, with the symmetron, of a
model subject to the Damour-Polyakov screening mechanism PhysRevD.82.063519 .
The experimental analysis in Pitschmann:2018aa  depends heavily on the field
profile of dilatons in both the one mirror or two mirror setups, where the
field is either present over an infinite plane of high density or is confined
between two such parallel planes. This is the purpose of this paper to provide
a detailed analysis of the environment-dependent dilaton in realistic
situations which can be tested using current experiments. For instance the
analysis of the LLR results as well as the screening of neutrons in Q-bounce
experiments employ the dilaton solution around and inside a sphere as derived
here. For a similar study applied to chameleon field theories see
Ivanov:2016rfs . The symmetron field equations have been studied in a similar
fashion in Brax:2017hna ; Pitschmann:2020ejb . Testing screened models can be
tackled from the laboratory to cosmological scales as reviewed in Brax:2021wcv
.

In section II we will provide some background information on dilatons, which
will provide the relevant definitions for the field theory analysis. We
consider a two-dimensional parameter subspace of the dilaton model, which is
of special cosmological significance as noted in section III. Then, in section
IV approximative solutions for the one mirror case will be derived, while in
section V the corresponding two mirror solutions will be given. In section VI
approximate dilaton solutions are derived for a spherical source. Section VII
provides relevant information on the qBOUNCE experiment, where the dilaton-
induced resonance frequency shift for the case of a single mirror has also
been summarized for a large range of parameters. In section VIII the induced
pressure in Casimir experiments due to the dilaton field between two mirrors
of the experimental setup is derived, while in section IX bounds due to LLR
are given. A conclusion in section X will be followed by Appendix A, in which
the precession of the lunar perigee induced by fifth forces is obtained.

##  II Background

Following Brax:2018iyo , the dilaton potential is given by

$\displaystyle V(\phi)=V_{0}\,e^{-\lambda\phi/m_{\text{Pl}}}\>,$ (1)

where $V_{0}$ is an energy scale related to the dark energy of the Universe
and $\lambda$ a numerical constant. This potential corresponds to the string
theory dilaton potential in the strong coupling limit Damour:1994zq ;
Damour:2002nv . Together with the coupling to matter this induces an effective
potential

$\displaystyle V_{\text{eff}}(\phi;\rho)=V(\phi)+A(\phi)\,\rho\>,$ (2)

where for the environment-dependent dilaton we have Brax:2010gi

$\displaystyle A(\phi)=1+\frac{A_{2}}{2m_{\text{Pl}}^{2}}\,\phi^{2}\>,$ (3)

and hence

$\displaystyle
V_{\text{eff}}(\phi;\rho)=V_{0}\,e^{-\lambda\phi/m_{\text{Pl}}}+\frac{A_{2}\rho}{2m_{\text{Pl}}^{2}}\,\phi^{2}\>.$
(4)

Here, we have neglected an additional term $\rho$, which does not affect the
equations of motion. The minimum value $\phi_{\rho}$ in the presence of a
density $\rho$ is given by
$V_{\text{eff},\phi}(\phi;\rho)\big{|}_{\phi=\phi_{\rho}}=0$ and reads

$\displaystyle\phi_{\rho}=\frac{m_{\text{Pl}}}{\lambda}\,W\bigg{(}\frac{\lambda^{2}V_{0}}{A_{2}\rho}\bigg{)}$
(5)

with the Lambert $W$-function

$\displaystyle W(x)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}\,x^{n}$
$\displaystyle=x-x^{2}+\frac{3}{2}\,x^{3}-\frac{8}{3}\,x^{4}+\frac{125}{24}\,x^{5}-\frac{54}{5}\,x^{6}+\ldots\>.$
(6)

For large arguments the approximative relation holds:

$\displaystyle W(x)\simeq\ln x\>.$ (7)

The mass $\mu_{\rho}$ of the quantum fluctuation is therefore

$\displaystyle\mu_{\rho}$
$\displaystyle=\sqrt{V_{\text{eff},\phi\phi}(\phi_{\rho};\rho)}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{\rho}/m_{\text{Pl}}}+A_{2}\rho}\>.$
(8)

We employ the metric signature ($+---$), for which the stress-energy tensor of
a scalar field is

$\displaystyle T_{\mu\nu}^{\phi}$
$\displaystyle=\partial_{\mu}\phi\,\partial_{\nu}\phi-
g_{\mu\nu}\Big{(}\frac{1}{2}\,\partial_{\alpha}\phi\,\partial^{\alpha}\phi-V(\phi;\rho)\Big{)}\>,$
(9)

while the equations of motion read

$\displaystyle\Box\phi+V_{\text{eff},\phi}(\phi;\rho)=0\>$ (10)

in the presence of matter.

##  III The ”Cosmological” Dilaton

The parameter space of the dilaton model is 3-dimensional ($V_{0}$, $\lambda$,
$A_{2}$). Since we are interested only in ”cosmological” dilatons, having
significance in the cosmological domain, we consider only the 2-dimensional
parameter subspace, for which

$\displaystyle V_{\text{eff}}(\phi_{V};\rho_{V})=3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}=6.73\times 10^{-34}\>\text{MeV}^{4}\>,$ (11)

where $\phi_{V}$, $\rho_{V}$ are the corresponding vacuum values and the
density parameter $\Omega_{\Lambda 0}\sim 0.73$. This choice neglects the
possible instability of the potential under radiative corrections. We take the
parameter $V_{0}$ as a function of $A_{2}$ and $\lambda$ such that Eq. (11) is
obeyed. It is important to notice that the coupling function involves the
strong coupling scale

$M=\frac{m_{\rm Pl}}{\sqrt{A}_{2}}.$ (12)

In the spirit of an effective field theory expansion of the coupling function,
one must require that $\phi/M\ll 1$ to guarantee that terms of higher order in
$\phi$ can be neglected. Similarly the exponential potential involves the
scale

$\Lambda=\frac{m_{\rm Pl}}{\lambda}$ (13)

and one must make sure that typically $\phi/\Lambda\gg 1$ to guarantee that
higher order string corrections in $e^{-n\phi/\Lambda}$ can be neglected. We
will make sure that these conditions are satisfied in the following.

It is straightforward to show that the condition (11) leads to

$\displaystyle V_{0}(A_{2},\lambda)$
$\displaystyle=\frac{A_{2}\rho_{V}}{\lambda^{2}}\left(\sqrt{1+\frac{2\lambda^{2}}{A_{2}\rho_{V}}\,3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}}-1\right)$
$\displaystyle\quad\times\exp\left\\{\sqrt{1+\frac{2\lambda^{2}}{A_{2}\rho_{V}}\,3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}}-1\right\\}\>.$ (14)

The dilaton parameter space becomes effectively 2-dimensional ($\lambda$,
$A_{2}$) simplifying also the representation of experimental constraints. For
illustrative purposes, in Fig. 1 the effective potential with its components
is depicted for two different sets of parameter values for $\lambda$ and
$A_{2}$.

Figure 1: The effective potential in Eq. (4) is plotted with its components
for two different sets of parameter values for $\lambda$ and $A_{2}$, while
the parameter $V_{0}$ varies as a function of $A_{2}$ and $\lambda$ such that
the effective potential of the minimum in vacuum equals $3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}$.

Using Eq. (III) in Eq. (II) the mass $\mu_{\rho}$ may be expressed as a
function of the parameters $\lambda$ and $A_{2}$ as well

$\displaystyle\mu_{\rho}$
$\displaystyle=\frac{\sqrt{A_{2}\rho}}{m_{\text{Pl}}}\,\Bigg{\\{}1+W\Bigg{[}\frac{\rho_{V}}{\rho}\left(\sqrt{1+\frac{2\lambda^{2}}{A_{2}\rho_{V}}\,3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}}-1\right)$
$\displaystyle\quad\times\exp\left(\sqrt{1+\frac{2\lambda^{2}}{A_{2}\rho_{V}}\,3\Omega_{\Lambda
0}m_{\text{Pl}}^{2}H_{0}^{2}}-1\right)\Bigg{]}\Bigg{\\}}^{1/2}\>.$ (15)

With these analytical expressions we can perform the limits $A_{2}\to\infty$,
$\lambda\to\infty$, $A_{2}\to 0$ or $\lambda\to 0$ and study the behavior
along curves $\lambda\propto\sqrt{A_{2}}$, which all provide some qualitative
insight of the relevant parameter region. This behavior is summarized in the
following two tables 1 and 2.

| $\phi_{\rho}$ | $\mu_{\rho}$
---|---|---
$\lim_{A_{2}\to\infty}$ | $0$ | $\infty$
$\lim_{\lambda\to\infty}$ | $\displaystyle m_{\text{Pl}}\,\sqrt{\frac{2}{A_{2}\rho_{V}}\,3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}}$ | $\infty$
$\lim_{A_{2}\to 0}$ | $\infty$ | $0$
$\lim_{\lambda\to 0}$ | $0$ | $\displaystyle\frac{\sqrt{A_{2}\rho}}{m_{\text{Pl}}}$
$\lambda\propto\sqrt{A_{2}}$ | $\sim 1/\lambda$ | $\sim\lambda$
Table 1: Here, Eqs. (5) and (II) are given for different limits. | $\displaystyle V_{0}\,e^{-\lambda\phi_{\rho}/m_{\text{Pl}}}$ | $\displaystyle\frac{A_{2}\rho}{2m_{\text{Pl}}^{2}}\,\phi_{\rho}^{2}$
---|---|---
$\lim_{A_{2}\to\infty}$ | $3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}$ | $0$
$\lim_{\lambda\to\infty}$ | $0$ | $\displaystyle\frac{\rho}{\rho_{V}}\,3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}$
$\lim_{A_{2}\to 0}$ | $0$ | $\displaystyle\frac{\rho}{\rho_{V}}\,3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}$
$\lim_{\lambda\to 0}$ | $3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}$ | $0$
$\lambda\propto\sqrt{A_{2}}$ | const | const
Table 2: The limits of the two constituent parts of the effective potential
Eq. (4) are summarized in this table.

We may summarize the findings of the tables as follows. For roughly
$\lambda\propto\sqrt{A_{2}}$ we expect finite limits to be obtainable from
experiments as long as $\lambda$ is not too large since otherwise
$\phi_{\rho}$ decreases, i.e. the dilaton effectively disappears and the
interaction range $1/\mu_{\rho}$ vanishes as well. For $A_{2}\gg\lambda$ we
expect no limits from experiments since $\phi_{\rho}$ becomes small. On the
other hand, for $\lambda\gg A_{2}$ we also do not expect bounds since in this
case either $1/\mu_{\rho}$ decreases without limit or $\phi_{\rho}$ diverges
and cannot act dynamically anymore. Hence, we find that the dilaton has a
physical impact only within a restricted region in the $A_{2}$, $\lambda$
parameter space. Consequently, all experimental constraints have to lie within
this region. As can be seen in table 1, shorter interaction ranges correspond
to larger values of $A_{2}$ and $\lambda$. Therefore, we expect LLR to probe
small parameter values within this region, table top experiments
correspondingly larger parameters and collider constraints still larger
values. Numerical evaluations in the accompanying article Pitschmann:2018aa
corroborate these findings. Notice that when $\lambda=\kappa\sqrt{A}_{2}$, the
validity of the model as an effective field theory is guaranteed when
$\kappa\ll 1$ as the field values must satisfy$\Lambda\ll\phi\ll M$ where
$\Lambda=\kappa M\ll M$. These expectations are also confirmed by the actual
bounds in Pitschmann:2018aa .

##  IV one mirror

In this section, we treat the case of a single mirror filling the infinite
half-space $z\leq 0$. The 1-dimensional equation of motion reads (see Eq.
(10))

$\displaystyle\frac{d^{2}\phi}{dz^{2}}=V_{\text{eff},\phi}(\phi;\rho)\>.$ (16)

Multiplication by $\phi^{\prime}$ and integration with respect to $z$ gives

$\displaystyle\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}-\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=z_{0}}=V_{\text{eff}}(\phi;\rho)-V_{\text{eff}}(\phi;\rho)\big{|}_{z=z_{0}}\>,$
(17)

with the integration constant $z_{0}$ and leads to

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\displaystyle\sqrt{V_{\text{eff}}(\phi;\rho)-V_{\text{eff}}(\phi;\rho)\big{|}_{z=z_{0}}+\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=z_{0}}}}$
$\displaystyle\qquad=\pm\sqrt{2}\left(z-z_{0}\right),$ (18)

where $\phi_{0}=\phi(z_{0})$, the $+$ sign is for $\phi^{\prime}\geq 0$, while
the $-$ sign holds for $\phi^{\prime}\leq 0$.

###  IV.1 The Vacuum Region

First, we consider the case of low density $\rho_{V}$ corresponding to the
medium above the mirror and search for a solution that asymptotically for
$z\to\infty$ goes as $\phi(z)\to\phi_{V}$ with

$\displaystyle\phi_{V}=\frac{m_{\text{Pl}}}{\lambda}\,W\bigg{(}\frac{\lambda^{2}V_{0}}{A_{2}\rho_{V}}\bigg{)}\>,$
(19)

implying $\phi^{\prime}\to 0$. We find for $z_{0}=0$ and $z\to\infty$ from Eq.
(17)

$\displaystyle-\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=0}=V_{\text{eff}}(\phi_{V};\rho_{V})-V_{\text{eff}}(\phi;\rho_{V})\big{|}_{z=0}\>.$
(20)

Using Eq. (20) in Eq. (IV) gives

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\displaystyle\sqrt{V_{\text{eff}}(\phi;\rho_{V})-V_{\text{eff}}(\phi_{V};\rho_{V})}}=\sqrt{2}\,z\>,$
(21)

respectively

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\sqrt{\displaystyle
V_{0}\left(e^{-\lambda\phi/m_{\text{Pl}}}-e^{-\lambda\phi_{V}/m_{\text{Pl}}}\right)+\frac{A_{2}\rho_{V}}{2m_{\text{Pl}}^{2}}\left(\phi^{2}-\phi_{V}^{2}\right)}}$
$\displaystyle=\sqrt{2}\,z\>.$ (22)

Approximating the effective potential around its minimum at the vacuum value
$\phi_{V}$, we find to leading order

$\displaystyle\frac{1}{\mu_{V}}\int_{(\phi_{V}-\phi(z))/m_{\text{Pl}}}^{(\phi_{V}-\phi_{0})/m_{\text{Pl}}}\frac{d\tilde{\phi}}{\tilde{\phi}}=z$
(23)

with the mass of the quantum fluctuation in vacuum

$\displaystyle\mu_{V}$
$\displaystyle=\sqrt{V_{\text{eff},\phi\phi}(\phi;\rho_{V})}\Big{|}_{\phi=\phi_{V}}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{V}/m_{\text{Pl}}}+A_{2}\rho_{V}}\>.$
(24)

Inverting the relation straightforwardly leads to

$\displaystyle\phi(z)$
$\displaystyle=\phi_{V}+\left(\phi_{0}-\phi_{V}\right)e^{-\mu_{V}z}\>.$ (25)

###  IV.2 The High Density Region

Here, we consider the case of high density $\rho_{M}$ as inside the mirror.
Clearly, for $z\to-\infty$ we have $\phi(z)\to\phi_{M}$ with

$\displaystyle\phi_{M}=\frac{m_{\text{Pl}}}{\lambda}\,W\bigg{(}\frac{\lambda^{2}V_{0}}{A_{2}\rho_{M}}\bigg{)}\>,$
(26)

and hence $\phi^{\prime}\to 0$. Therefore, we find for $z_{0}=0$ and
$z\to-\infty$ from Eq. (17)

$\displaystyle-\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=0}=V_{\text{eff}}(\phi_{M};\rho_{M})-V_{\text{eff}}(\phi;\rho_{M})\big{|}_{z=0}\>.$
(27)

Using Eq. (27) in Eq. (IV) gives

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\displaystyle\sqrt{V_{\text{eff}}(\phi;\rho_{M})-V_{\text{eff}}(\phi_{M};\rho_{M})}}=\sqrt{2}\,z\>.$
(28)

Proceeding analogously to the case of vacuum, we finally obtain

$\displaystyle\phi(z)$
$\displaystyle=\phi_{M}+\left(\phi_{0}-\phi_{M}\right)e^{\mu_{M}z}\>,$ (29)

with the mass of the quantum fluctuation in the mirror

$\displaystyle\mu_{M}=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{M}/m_{\text{Pl}}}+A_{2}\rho_{M}}\>.$
(30)

###  IV.3 Boundary Conditions

Using the boundary conditions

$\displaystyle\frac{d\phi}{dz}\bigg{|}_{z=0_{-}}=\frac{d\phi}{dz}\bigg{|}_{z=0_{+}}\>,$
(31)

we find

$\displaystyle\phi_{0}=\frac{\mu_{V}\,\phi_{V}+\mu_{M}\,\phi_{M}}{\mu_{V}+\mu_{M}}\>.$
(32)

The second boundary condition

$\displaystyle\phi(0_{-})=\phi(0_{+})\>,$ (33)

is trivially satisfied.

###  IV.4 Final Solution

Summarising, we obtain the solution

$\displaystyle\phi(z)$
$\displaystyle=\Theta(+z)\,\big{(}\phi_{V}+\left(\phi_{0}-\phi_{V}\right)e^{-\mu_{V}z}\big{)}$
$\displaystyle\quad+\Theta(-z)\,\big{(}\phi_{M}+\left(\phi_{0}-\phi_{M}\right)e^{\mu_{M}z}\big{)}\>,$
(34)

with

$\displaystyle\mu_{V}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{V}/m_{\text{Pl}}}+A_{2}\rho_{V}}\>,$
$\displaystyle\mu_{M}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{M}/m_{\text{Pl}}}+A_{2}\rho_{M}}\>,$
(35)

and

$\displaystyle\phi_{0}=\frac{\mu_{V}\,\phi_{V}+\mu_{M}\,\phi_{M}}{\mu_{V}+\mu_{M}}\>.$
(36)

A prototype solution with values $A_{2}=10^{35}$, $\lambda=10^{25}$ and
$V_{0}=1.00034\times 3\Omega_{\Lambda 0}m_{\text{Pl}}^{2}H_{0}^{2}=6.73\times
10^{-10}$ meV4, $\rho_{V}=10^{-15}$ MeV4 and $\rho_{M}=1.082\times 10^{-5}$
MeV4 is plotted in Fig. 2.

Figure 2: This plot shows the dilaton solution with the blue area depicting
the mirror of the experimental setup. We can see that $\lambda\phi(z)\ll
m_{\text{Pl}}=1.22\times 10^{22}$ MeV is indeed satisfied in this case.

These dilaton solutions will be instrumental in analyzing the results of
qBOUNCE experiments as outlined in section VII.

##  V two mirrors

In this section, we treat the case of two parallel infinitely thick mirrors
separated at distance $2d$ in $z$-direction with $z=0$ being the center
between the two mirrors.

###  V.1 The Vacuum Region

Here, we consider the case of low density $\rho_{V}$, as between the mirrors
and choose $z_{0}=0$. Due to the symmetry of the setup, the derivative of the
field has to vanish there yielding for Eq. (IV)

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\displaystyle\sqrt{V_{\text{eff}}(\phi;\rho_{V})-V_{\text{eff}}(\phi_{0};\rho_{V})}}=\sqrt{2}\,z\>,$
(37)

where $\phi_{0}=\phi(0)$. Approximating the potential around $\phi_{0}$ yields

$\displaystyle
V_{\text{eff}}(\phi;\rho_{V})-V_{\text{eff}}(\phi_{0};\rho_{V})\simeq-\mathfrak{D}_{0}\left(\phi-\phi_{0}\right)+\frac{\mu_{0}^{2}}{2}\left(\phi-\phi_{0}\right)^{2}\>,$
(38)

with the positive constants

$\displaystyle\mathfrak{D}_{0}$ $\displaystyle=\frac{\lambda
V_{0}}{m_{\text{Pl}}}\,e^{-\lambda\phi_{0}/m_{\text{Pl}}}-\frac{A_{2}\rho_{V}}{m_{\text{Pl}}^{2}}\,\phi_{0}\>,$
$\displaystyle\mu_{0}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{0}/m_{\text{Pl}}}+A_{2}\rho_{V}}\>,$
(39)

leads to

$\displaystyle\int_{\phi_{0}}^{\phi(z)}\frac{d\phi}{\sqrt{\displaystyle-\mathfrak{D}_{0}\left(\phi-\phi_{0}\right)+\frac{\mu_{0}^{2}}{2}\left(\phi-\phi_{0}\right)^{2}}}=-\sqrt{2}\,|z|\>.$
(40)

Substituting $\displaystyle
x=1+\frac{\mu_{0}^{2}}{\mathfrak{D}_{0}}\left(\phi_{0}-\phi\right)$ leads to
the elementary integral

$\displaystyle\frac{1}{\mu_{0}}\int_{1}^{1+\frac{\mu_{0}^{2}}{\mathfrak{D}_{0}}\,(\phi_{0}-\phi(z))}\frac{dx}{\sqrt{\displaystyle
x^{2}-1}}=|z|\>.$ (41)

Integration and some transformations lead to the solution

$\displaystyle\phi(z)=\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\,\big{(}1-\cosh(\mu_{0}z)\big{)}\>.$
(42)

###  V.2 The High Density Region

We can read off the solution inside the mirrors directly from the
corresponding solution in the one mirror case Eq. (IV.4)

$\displaystyle\phi(z)$
$\displaystyle=\phi_{M}+\left(\phi_{d}-\phi_{M}\right)e^{-\mu_{M}(|z|-d)}\>,$
(43)

where $\phi_{d}=\phi(d)=\phi(-d)$ is the field value at the surfaces of the
mirrors.

###  V.3 Boundary Conditions

Using the boundary conditions at the mirror surface

$\displaystyle\phi(d_{-})=\phi(d_{+})\>,$ (44)

gives $\phi_{d}$ as a function of $\phi_{0}$

$\displaystyle\phi_{d}=\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\left(1-\cosh(\mu_{0}d)\right)\>,$
(45)

which together with the second boundary condition

$\displaystyle\frac{d\phi}{dz}\bigg{|}_{z=d_{-}}=\frac{d\phi}{dz}\bigg{|}_{z=d_{+}}$
(46)

gives an equation, which defines $\phi_{0}$ implicitly

$\displaystyle\phi_{M}+\frac{\mathfrak{D}_{0}}{\mu_{0}\mu_{M}}\sinh(\mu_{0}d)=\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\left(1-\cosh(\mu_{0}d)\right)$
(47)

since $\mathfrak{D}_{0}$ and $\mu_{0}$ both are functions of $\phi_{0}$.

###  V.4 Final Solution

Summarising, we obtain the solution

$\displaystyle\phi(z)$
$\displaystyle=\Theta(d-|z|)\left\\{\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\,\big{(}1-\cosh(\mu_{0}z)\big{)}\right\\}$
$\displaystyle\quad+\Theta(|z|-d)\left\\{\phi_{M}+\left(\phi_{d}-\phi_{M}\right)e^{-\mu_{M}(|z|-d)}\right\\}\>,$
(48)

where

$\displaystyle\mu_{M}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{M}/m_{\text{Pl}}}+A_{2}\rho_{M}}\>,$
$\displaystyle\mu_{0}$
$\displaystyle=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{0}/m_{\text{Pl}}}+A_{2}\rho_{V}}\>,$
$\displaystyle\mathfrak{D}_{0}$ $\displaystyle=\frac{\lambda
V_{0}}{m_{\text{Pl}}}\,e^{-\lambda\phi_{0}/m_{\text{Pl}}}-\frac{A_{2}\rho_{V}}{m_{\text{Pl}}^{2}}\,\phi_{0}\>,$
(49)

$\phi_{d}$ as a function of $\phi_{0}$ is given by

$\displaystyle\phi_{d}=\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\left(1-\cosh(\mu_{0}d)\right)\>,$
(50)

and $\phi_{0}$ is solution to

$\displaystyle\phi_{M}+\frac{\mathfrak{D}_{0}}{\mu_{0}\mu_{M}}\sinh(\mu_{0}d)=\phi_{0}+\frac{\mathfrak{D}_{0}}{\mu_{0}^{2}}\left(1-\cosh(\mu_{0}d)\right)\>.$
(51)

A prototype solution is plotted in Fig. 3. The induced pressure inside the
planes is $\displaystyle P=-4.73\times 10^{-4}$ pN/cm2 (see section VII).

Figure 3: This plot shows the dilaton solution with the blue and orange area
depicting the mirrors of the experimental setup. We can see that
$\lambda\phi(z)\ll m_{\text{Pl}}=1.22\times 10^{22}$ MeV is indeed satisfied
in this case.

These dilaton solutions will be instrumental in analyzing the induced pressure
in Casimir experiments.

##  VI Dilaton Field of a Sphere

In this section we consider the dilaton field as caused by a static massive
sphere surrounded by vacuum with radius $R$ and a homogenous high density
$\rho_{S}$, which is used for the analysis of the qBOUNCE and Lunar Laser
Ranging bounds.

Since neutrons are used in the search for dilatons in qBOUNCE experiments and
neutron interferometry, it is important to understand their interaction with
the dilaton. For a certain parameter regime this interaction between neutron
and dilaton becomes strong. In this case, the neutron affects the background
dilaton field as generated by the mirrors of the experimental setup in a non-
negligible way, which in turn weakens the effect on the neutron, viz.
screening of the neutron sets in (see also Burrage_2021 ). Since we treat the
dilaton as a classical field theory, a consistent description of its coupling
to a quantum mechanical system is beyond our reach. Therefore, we employ a
semi-classical treatment in which the neutron’s probability distribution times
its mass acts as the source of the dilaton as defined in Eq. (81).

Concerning Lunar Laser Ranging, we approximate the Sun, Earth and Moon as
spherical sources of dilatons with homogenous density and employ the solutions
derived in this section.

The spherically symmetric field equation is given by

$\displaystyle\frac{d^{2}\phi}{dr^{2}}+\frac{2}{r}\frac{d\phi}{dr}$
$\displaystyle=V_{\text{eff},\phi}(\phi;\rho)\>,$ (52)

with the boundary conditions

$\displaystyle\phi^{\prime}(0)$ $\displaystyle=0\>,$
$\displaystyle\lim_{r\to\infty}\phi(r)$ $\displaystyle\to\phi_{V}\>.$ (53)

Inside the sphere we have density $\rho_{S}$. The corresponding minimum of the
field value is

$\displaystyle\phi_{S}=\frac{m_{\text{Pl}}}{\lambda}\,W\bigg{(}\frac{\lambda^{2}V_{0}}{A_{2}\rho_{S}}\bigg{)}\>.$
(54)

We expand

$\displaystyle V_{\text{eff},\phi}(\phi;\rho_{S})$ $\displaystyle\simeq
V_{\text{eff},\phi\phi}(\phi_{S};\rho_{S})\left(\phi-\phi_{S}\right)$
$\displaystyle=\mu_{S}^{2}\left(\phi-\phi_{S}\right)$ (55)

with

$\displaystyle\mu_{S}=\frac{1}{m_{\text{Pl}}}\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{S}/m_{\text{Pl}}}+A_{2}\rho_{S}}\>.$
(56)

For the field equation we find

$\displaystyle\frac{d^{2}\phi}{dr^{2}}+\frac{2}{r}\frac{d\phi}{dr}\simeq\mu_{S}^{2}\left(\phi-\phi_{S}\right).$
(57)

Introducing the field $\varphi$

$\displaystyle\phi-\phi_{S}=\frac{\varphi}{r}\>,$ (58)

Eq. (57) takes on the simpler form

$\displaystyle\frac{d^{2}\varphi}{dr^{2}}\simeq\mu_{S}^{2}\,\varphi\>.$ (59)

From the general solution for $\varphi$ we obtain the particular solution for
$\phi$, which is convergent for $r\to 0$ and satisfies the boundary condition
$\phi^{\prime}(0)=0$:

$\displaystyle\phi(r)=\phi_{S}+\mathfrak{C}_{I}\frac{\displaystyle\sinh(\mu_{S}r)}{r}\>,$
(60)

where $\mathfrak{C}_{I}$ is some constant to be determined later.

In the vacuum outside the sphere we approximate

$\displaystyle V_{\text{eff},\phi}(\phi;\rho_{V})$ $\displaystyle\simeq
V_{\text{eff},\phi\phi}(\phi_{V};\rho_{V})\left(\phi-\phi_{V}\right)$
$\displaystyle=\mu_{V}^{2}\left(\phi-\phi_{V}\right),$ (61)

where $\mu_{V}$ is given by Eq. (IV.1).

For the field equation we find

$\displaystyle\frac{d^{2}\phi}{dr^{2}}+\frac{2}{r}\frac{d\phi}{dr}\simeq\mu_{V}^{2}\left(\phi-\phi_{V}\right).$
(62)

With the field $\varphi$

$\displaystyle\phi-\phi_{V}=\frac{\varphi}{r}\>,$ (63)

the field equation again simplifies to

$\displaystyle\frac{d^{2}\varphi}{dr^{2}}\simeq\mu_{V}^{2}\,\varphi\>.$ (64)

The general solution of which provides the $\phi$ solution convergent for
$r\to\infty$ and satisfying the boundary condition
$\displaystyle\lim_{r\to\infty}\phi(r)\to\phi_{V}$

$\displaystyle\phi(r)=\phi_{V}+\mathfrak{C}_{O}\frac{e^{-\mu_{V}(r-R)}}{r}\>,$
(65)

where $\mathfrak{C}_{O}$ is some constant to be determined next.

The dilaton field has to satisfy the following boundary conditions at the
surface of the sphere

$\displaystyle\phi(R_{-})$ $\displaystyle=\phi(R_{+})\>,$
$\displaystyle\phi^{\prime}(R_{-})$ $\displaystyle=\phi^{\prime}(R_{+})\>,$
(66)

which provides explicit expressions for $\mathfrak{C}_{I}$ and
$\mathfrak{C}_{O}$. After some elementary algebraical manipulations we obtain
the relations

$\displaystyle\mathfrak{C}_{I}$
$\displaystyle=\frac{\phi_{V}-\phi_{S}}{\cosh(\mu_{S}R)}\frac{1+\mu_{V}R}{\mu_{S}+\mu_{V}\tanh(\mu_{S}R)}\>,$
$\displaystyle\mathfrak{C}_{O}$
$\displaystyle=-R\,\frac{1-\frac{1}{\mu_{S}R}\tanh(\mu_{S}R)}{1+\frac{\mu_{V}}{\mu_{S}}\tanh(\mu_{S}R)}\left(\phi_{V}-\phi_{S}\right)\>.$
(67)

In order to quantify the amount of ”screening” of a sphere, we introduce a
formfactor, which we interpret as a ”screening charge” $\mathfrak{Q}$. The
interaction of the sphere with its surrounding is given by its outer field.
Clearly, with decreasing radius $R$ the sphere becomes increasingly
”unscreened” since the dilaton field inside the sphere decreases towards its
minimum throughout the volume of the sphere. On the other hand, for large
sphere radii the dilaton field reaches its minimum already before the center
of the sphere, it is no longer ”sourced” by the whole volume of the sphere,
which becomes ”screened”.

An unscreened sphere has $R\ll 1/\mu_{S}$. Hence, we expand $\mathfrak{C}_{O}$
in powers of $\mu_{S}R$ to obtain for an ”unscreened” sphere

$\displaystyle\mathfrak{C}_{O}^{u}=-\frac{\mu_{S}^{2}R^{3}}{3}\left(\phi_{V}-\phi_{S}\right)\>.$
(68)

We define the ”screening charge” $\mathfrak{Q}$ as the ratio

$\displaystyle\mathfrak{Q}=\frac{\mathfrak{C}_{O}}{\mathfrak{C}_{O}^{u}}=\frac{3}{\mu_{S}^{2}R^{2}}\frac{1-\frac{1}{\mu_{S}R}\tanh(\mu_{S}R)}{1+\frac{\mu_{V}}{\mu_{S}}\tanh(\mu_{S}R)}\>.$
(69)

With this definition, see Fig. 4,

$\displaystyle\mathfrak{Q}\to\begin{dcases}0\>,&\quad\text{for "screened"
bodies with }\mu_{S}R\to\infty\>,\\\ 1\>,&\quad\text{for "unscreened" bodies
with }\mu_{S}R\to 0\>.\end{dcases}$

Figure 4: Top: The ”screening charge” $\mathfrak{Q}$ is plotted as a function
of the radius $R$ of a sphere. The parameters taken are $\rho_{N}=1.44\times
10^{10}$ MeV4, the density of a neutron, $A_{2}=10^{40}$, $\lambda=10^{15}$,
$V_{0}$ according to Eq. (III) and $R$ varying between $0$ and $3\times
R_{N}=0.0075$ MeV-1, where $R_{N}$ is the radius of a neutron. Bottom: The
field profile of a sphere (neutron) is plotted as a function of the radial
distance of the center of the sphere with parameters $A_{2}=10^{39}$,
$\lambda=10^{25}$ and $V_{0}$ according to Eq. (III). The blue line
corresponds to the density of a neutron $\rho_{N}=1.44\times 10^{10}$ MeV4
providing a ”screening charge” $\mathfrak{Q}=0.806$ and the yellow line to
$10\times\rho_{N}$ providing $\mathfrak{Q}=0.298$. The blue square is bounded
by the radius $R_{N}$ of the sphere (neutron) and the vacuum field value
$\phi=\phi_{V}$.

Finally, we find for the solution

$\displaystyle\phi(r)=\begin{dcases}&\phi_{S}+\frac{\phi_{V}-\phi_{S}}{\cosh(\mu_{S}R)}\frac{1+\mu_{V}R}{\mu_{S}+\mu_{V}\tanh(\mu_{S}R)}\\\
&\hskip 25.60747pt\times\>\frac{\sinh(\mu_{S}r)}{r}\>,\hskip
75.3998pt\text{for }r\leq R\>,\\\
&\phi_{V}-\mathfrak{Q}\,\frac{\mu_{S}^{2}R^{3}}{3}\left(\phi_{V}-\phi_{S}\right)\frac{e^{-\mu_{V}(r-R)}}{r}\>,\text{for
}r\geq R\>.\end{dcases}$

The acceleration experienced by a pointlike test body, which does not disturb
the field, in the outer field of the sphere has been derived in
Pitschmann:2020ejb  for a generic scalar $\phi$ in the non-relativistic limit
as

$\displaystyle\vec{f}_{\phi}=-m\vec{\nabla}\ln A(\phi)\>.$ (70)

In all models of consideration $A(\phi)=1+\delta A(\phi)$ with $\delta
A(\phi)\ll 1$, such that $\ln A(\phi)\simeq\delta A(\phi)$. Consequently, we
find for the force on a particle caused by a scalar $\phi$ to leading order

$\displaystyle\vec{f}_{\phi}$ $\displaystyle=-m\vec{\nabla}A(\phi)\>.$ (71)

In the case of dilatons Eq. (3) holds and we finally obtain for the dilaton
force on a point particle

$\displaystyle\vec{f}_{\phi}$
$\displaystyle=-m\,\frac{A_{2}}{m_{\text{Pl}}^{2}}\,\phi\,\vec{\nabla}\phi\>,$
(72)

and, respectively, for the acceleration

$\displaystyle\vec{a}_{\phi}$
$\displaystyle=-\frac{A_{2}}{m_{\text{Pl}}^{2}}\,\phi\,\vec{\nabla}\phi\>.$
(73)

Asymptotically, for large $r$ we obtain

$\displaystyle\vec{a}_{\phi}\to-\mathfrak{Q}\,\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{S}^{2}R^{3}}{3}\,\phi_{V}\left(\phi_{V}-\phi_{S}\right)\frac{e^{-\mu_{V}(r-R)}}{r}\frac{\vec{r}}{r}\>,$
(74)

justifying the definition of $\mathfrak{Q}$ as a ”screening charge”.

These dilaton solutions will be instrumental in analyzing the results of
qBOUNCE experiment and Lunar Laser Ranging in the following sections.

##  VII Dilaton-induced Frequency Shift in qBOUNCE

Here, we derive a discrete set of limits for the qBOUNCE experiment
Abele:2009dw ; Jenke:2011zz ; Jenke:2014yel  using the solutions obtained
herein. In this experiment, ultra-cold neutrons are dropped in Earth’s
gravitational potential and reflected by a neutron mirror, which has been
reported for the first time in Nesvizhevsky:2002ef . The energy eigenstates
are discrete and allow to apply the method of resonance spectroscopy. In
Jenke:2011zz  the basic setup is described. In the Rabi version of the
experiment, spectroscopy has been realized with energy resolution
3$\times$10-15 peV Cronenberg:2018qxf .

The experimental setup is such, that ultracold neutrons pass three regions,
while being reflected on polished glass mirrors. In Cronenberg:2018qxf , the
resonance spectroscopy transitions between the energy ground state
$E_{1}=1.41$ peV and the excited states $E_{3}=3.32$ peV as well as
$E_{4}=4.08$ peV have been demonstrated. First, the neutrons pass region I
which acts as a state selector for the ground state $|1\rangle$ having energy
$E_{1}$. A polished mirror at the bottom and a rough absorbing scatterer on
top at a height of about 20 $\mu$m serve to select the ground state. Neutrons
in higher, unwanted states are scattered out of the system. This region has a
length of 15 cm. Subsequently, in region II, a horizontal mirror performs
harmonic oscillations with a tunable frequency $\omega$, which drives the
system into a coherent superposition of ground and excited states. The length
of this region is 20 cm. Finally, region III is identical to the first region
and hence acts again as a ground state selector.

The quantum mechanical description of a neutron above a mirror in the
gravitational potential is given by the Schrödinger equation Westphal:2006dj .
After separation into free transversal and bound vertical states

$\displaystyle\Psi_{n}^{(0)}(\textbf{x},t)=\frac{e^{\frac{i}{\hbar}(p_{\perp}\cdot
x_{\perp}-E_{\perp}t)}}{2\pi\hbar
v_{\perp}}\,\psi_{n}^{(0)}(z)\,e^{-\frac{i}{\hbar}E_{n}t}\>,$ (75)

it reads

$\displaystyle-\frac{\hbar^{2}}{2m_{N}}\frac{\partial^{2}\psi_{n}^{(0)}(z)}{\partial
z^{2}}+m_{N}gz\,\psi_{n}(z)=E_{n}\psi_{n}^{(0)}(z)\>.$ (76)

The characteristic length scale

$\displaystyle
z_{0}=\sqrt[3]{\frac{\hbar^{2}}{2m_{N}^{2}g}}=5.87\,\mu\text{m}\>,$ (77)

and energy scale $E_{0}$ = $\sqrt[3]{\hbar^{2}m_{N}g^{2}/2}$ are given by the
mass $m_{N}$ of the neutron and the acceleration of the Earth $g$. With the
substitution

$\displaystyle\sigma=\sqrt[3]{\frac{2m_{N}^{2}g}{\hbar^{2}}}\left(z-\frac{E_{n}}{m_{N}g}\right)\equiv\frac{z-z_{n}}{z_{0}}\>,$
(78)

Eq. (76) becomes Airy’s equation

$\displaystyle\frac{d^{2}\tilde{\psi}_{n}(\sigma)}{d\sigma^{2}}-\sigma\,\tilde{\psi}_{n}(\sigma)=0\>.$
(79)

From the effective dilaton potential

$\displaystyle
V_{\text{eff}}(\phi)=V_{0}\,e^{-\lambda\phi/m_{\text{Pl}}}+\frac{A_{2}\rho}{2m_{\text{Pl}}^{2}}\,\phi^{2}\>,$
(80)

we can deduce the semi-classical neutron-dilaton coupling

$\displaystyle
V_{\text{eff}}=\frac{A_{2}}{2}\frac{m_{N}}{m_{\text{Pl}}^{2}}\,\psi^{*}\psi\,\phi^{2}\>.$
(81)

There are some subtleties involved here of a nature similar as in the case of
the symmetron. We refer to Brax:2017hna  for further information. The
corresponding quantum mechanical perturbation potential is given by

$\displaystyle\textrm{V}=\frac{A_{2}}{2}\frac{m_{N}}{m_{\text{Pl}}^{2}}\,\phi^{2}\>,$
(82)

and leads to a resonance frequency shift to first order (see e.g.
landau1991quantenmechanik ):

$\displaystyle\delta E_{mn}^{(1)}$ $\displaystyle\equiv
E_{m}^{(1)}-E_{n}^{(1)}$
$\displaystyle=\frac{A_{2}}{2}\frac{m_{N}}{m_{\text{Pl}}^{2}}\int_{-\infty}^{\infty}dz\,\Big{(}\big{|}\psi_{m}^{(0)}(z)\big{|}^{2}-\big{|}\psi_{n}^{(0)}(z)\big{|}^{2}\Big{)}\,\phi^{2}(z)\>.$
(83)

Likewise, the first order correction to the wavefunctions reads (see e.g.
landau1991quantenmechanik )

$\displaystyle\Psi_{n}^{(1)}(\textbf{x},t)=\sum_{m\neq
n}\frac{\displaystyle\int
d^{3}x^{\prime}\,\psi_{m}^{(0)*}(\textbf{x}^{\prime})\textrm{V}\psi_{n}^{(0)}(\textbf{x}^{\prime})}{E_{n}^{(0)}-E_{m}^{(0)}}\,\psi_{m}^{(0)}(\textbf{x})\,e^{-\frac{i}{\hbar}E_{n}t}\>.$
(84)

Hence, the correction to the density
$\displaystyle\varrho_{n}^{(0)}(z)=\psi_{n}^{(0)*}(z)\psi_{n}^{(0)}(z)$ to
first order is given by

$\displaystyle\varrho_{n}^{(1)}(z)$
$\displaystyle=2\,\mathfrak{Re}\big{(}\psi_{n}^{(0)*}(z)\psi_{n}^{(1)}(z)\big{)}$
$\displaystyle=A_{2}\,\frac{m_{N}}{m_{\text{Pl}}^{2}}\,\mathfrak{Re}\sum_{m\neq
n}\frac{\displaystyle\int_{-\infty}^{\infty}dz^{\prime}\,\psi_{m}^{(0)*}(z^{\prime})\,\psi_{n}^{(0)}(z^{\prime})\,\phi(z^{\prime})^{2}}{E_{n}^{(0)}-E_{m}^{(0)}}$
$\displaystyle\quad\times\psi_{n}^{(0)*}(z)\psi_{m}^{(0)}(z)\>,$ (85)

where $\mathfrak{Re}$ denotes the real part.

In the one mirror case the unperturbed normalized wavefunction for $z>0$ reads
(see e.g. Pitschmann:2019boa )

$\displaystyle\psi_{n}^{(0)}(z)=\frac{\text{Ai}\Big{(}\displaystyle\frac{z-z_{n}}{z_{0}}\Big{)}}{\sqrt{z_{0}}\,\text{Ai}^{\prime}\Big{(}-\displaystyle\frac{z_{n}}{z_{0}}\Big{)}}$
(86)

with $z_{n}=E_{n}/(mg)$. Outside this region the wavefunction vanishes. For a
single mirror filling the region $z\leq 0$ we can use Eq. (IV.4) and obtain
for the resonance frequency shift

$\displaystyle\delta E_{mn}^{(1)}$
$\displaystyle=\frac{A_{2}}{2}\frac{m_{N}}{m_{\text{Pl}}^{2}}\frac{1}{z_{0}}\int_{0}^{\infty}dz\,\Big{(}\phi_{V}+\left(\phi_{0}-\phi_{V}\right)e^{-\mu_{V}z}\Big{)}^{2}$
$\displaystyle\quad\times\Bigg{\\{}\frac{\displaystyle\text{Ai}\Big{(}\frac{z-z_{m}}{z_{0}}\Big{)}^{2}}{\text{Ai}^{\prime}\Big{(}-\displaystyle\frac{z_{m}}{z_{0}}\Big{)}^{2}}-\frac{\displaystyle\text{Ai}\Big{(}\frac{z-z_{n}}{z_{0}}\Big{)}^{2}}{\text{Ai}^{\prime}\Big{(}-\displaystyle\frac{z_{n}}{z_{0}}\Big{)}^{2}}\Bigg{\\}}\>.$
(87)

It is straightforward to find all the corresponding expressions in the two
mirror case. These expressions are very elaborate in their full detail and
hence we will refrain from reproducing them herein.

In order to account for the ”screening” of the neutron itself for the
extraction of the experimental limits, the transition energies should be
replaced as follows

$\displaystyle\delta E_{pq}\to\mathfrak{Q}\,\delta E_{pq}\>,$ (88)

where $\mathfrak{Q}$ is given in Eq. (69). The whole parameter space of
dilatons can be efficiently constrained using the results obtained herein. The
corresponding analysis has been carried out in parallel to this work
Pitschmann:2018aa , where the solutions obtained here for a one mirror setup
are used to exclude regions of the dilaton parameter space.

Nevertheless, in Table 3 we summarize the resonance frequency shift for a
range of dilaton parameters near the experimental sensitivity in the case of a
single mirror.

$\delta E_{14}^{(1)}$ [eV] | $A_{2}$ | $\lambda$
---|---|---
$3.03121\times 10^{-21}$ | $10^{35}$ | $10^{25}$
$5.71509\times 10^{-16}$ | $10^{35}$ | $10^{30}$
$5.75507\times 10^{-21}$ | $10^{35}$ | $10^{35}$
$2.60699\times 10^{-21}$ | $10^{40}$ | $10^{25}$
$6.76315\times 10^{-12}$ | $10^{40}$ | $10^{30}$
$5.59563\times 10^{-16}$ | $10^{40}$ | $10^{35}$
$1.51542\times 10^{-24}$ | $10^{45}$ | $10^{25}$
$1.51550\times 10^{-14}$ | $10^{45}$ | $10^{30}$
$3.02715\times 10^{-13}$ | $10^{45}$ | $10^{35}$
Table 3: $\delta E_{41}^{(1)}$ for $1$ mirror, viz. Eq. (VII), for the value
$\rho=2.51$ g/cm3.

As can be seen in this table, the deviation can be larger than the
experimental uncertainty of order $3\times 10^{-15}$ eV for cases where for
instance $A_{2}=10^{40}$ and $\lambda=10^{30}$ corresponding to $M\simeq
10^{-2}$ GeV and $\Lambda\simeq 10^{-3}$ eV, i.e. the dark energy scale. As a
result the forthcoming results in Pitschmann:2018aa  will test an interesting
range of scales for both particle physics and cosmology.

##  VIII Dilaton-induced Pressure in CASIMIR experiments

Here, we consider limits that can be obtained by the Casimir And Non-Newtonian
force EXperiment (Cannex) Sedmik:2018kqt  (see also Klimchitskaya:2019fsm ;
Klimchitskaya:2019nzu ). This experiment consists of two parallel plates in a
vacuum chamber and has been devised to measure the Casimir force and
hypothetical fifth forces. A dilaton field would induce a pressure between
those plates, which can be measured with high precision.

We approximate the setup in one dimension along the $z$-axis as follows.
Between the upper surface of the fixed lower mirror at $z=0$ and the lower
surface of the movable upper mirror at $z=d$ there is vacuum. Then follows the
upper mirror with thickness $D$ and above that at $z>d+D$ there is vacuum
again. In order to obtain the induced pressure for the Cannex setup, we can
apply the force induced on a point particle by dilatons in Eq. (71). We obtain

$\displaystyle\vec{f}_{\phi}=-\rho_{M}\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy\int_{d}^{d+D}dz\,\partial_{z}\ln
A(\phi)\,\vec{e}_{z}\>.$ (89)

Consequently, the pressure in $z$-direction is given by

$\displaystyle P_{z}=-\rho_{M}\int_{d}^{d+D}dz\,\partial_{z}\ln A(\phi)\>.$
(90)

The corresponding integral is a surface term and hence trivially carried out
with the final result

$\displaystyle P_{z}=\rho_{M}\,\Big{(}\ln A\big{(}\phi(d)\big{)}-\ln
A\big{(}\phi(d+D)\big{)}\Big{)}\>.$ (91)

This agrees with (see e.g. Pitschmann:2020ejb )

$\displaystyle\nabla^{\mu}T_{\mu\nu}=\partial_{\nu}\ln A\,T\>,$ (92)

where $T=g^{\mu\nu}T_{\mu\nu}$ and which reduces for a static field
configuration of $\phi$ to

$\displaystyle\partial^{z}T_{zz}=-\partial_{z}P_{z}=\partial_{z}\ln
A\,\rho_{M}\>.$ (93)

In all models of consideration $A(\phi)=1+\delta A(\phi)$ with $\delta
A(\phi)\ll 1$, such that $\ln A(\phi)\simeq\delta A(\phi)$. For dilatons Eq.
(3) holds and we finally obtain the dilaton-induced pressure

$\displaystyle
P_{z}=\rho_{M}\,\frac{A_{2}}{2m_{\text{Pl}}^{2}}\left(\phi^{2}(d)-\phi^{2}(d+D)\right).$
(94)

For $\phi(d)$ we employ the value $\phi_{d}$ at the mirror surface of the
corresponding two mirror solution given in Eq. (50), while for $\phi(d+D)$ we
can use the value $\phi_{0}$ at the mirror surface of the one mirror solution
given in Eq. (36) instead. Finally, we obtain for the pressure

$\displaystyle
P_{z}=\rho_{M}\,\frac{A_{2}}{2m_{\text{Pl}}^{2}}\left\\{\phi_{d}^{2}-\left(\frac{\mu_{V}\,\phi_{V}+\mu_{M}\,\phi_{M}}{\mu_{V}+\mu_{M}}\right)^{2}\right\\}.$
(95)

This results can be used to carry out a numerical analysis Pitschmann:2018aa
where the exclusion regions of the dilaton parameter space are obtained from
the Cannex experiment as we expect a sensitivity of $|P_{z}|\leq 1$ nPa.

##  IX Dilaton Constraints by Lunar Laser Ranging

In this section we analyze bounds on dilatons due to Lunar Laser Ranging
(LLR), which provides three separate constraints nordtvedt2001lunar  due to

  1. 1.

the Nordvedt effect, which relates to a difference between the free fall
acceleration of the Earth and Moon towards the Sun (equivalence principle
violations),

  2. 2.

deviations from the inverse-square law at distances comparable to the Earth-
Moon separation and

  3. 3.

time-variation of $G$.

Since the dilaton interaction is time-independent, it cannot induce a time-
variation of the effective coupling $G$. Hence, only the first two constraints
can lead to dilaton bounds and will be analyzed here.

###  IX.1 Constraints due to the Nordvedt Effect

For the first LLR test we consider the acceleration of the Earth (${\earth}$)
and Moon (${\leftmoon}$) towards the Sun (${\astrosun}$). For the Earth we
find 74

$\displaystyle\vec{a}_{\earth}$
$\displaystyle=\vec{a}_{G}+\vec{a}_{\phi{\earth}}$
$\displaystyle\simeq\vec{a}_{G}-\mathfrak{Q}_{\astrosun}\mathfrak{Q}_{\earth}\,\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{\astrosun}^{2}R_{\astrosun}^{3}}{3}\,\phi_{V}\left(\phi_{V}-\phi_{\astrosun}\right)$
$\displaystyle\quad\times\frac{e^{-\mu_{V}(r_{AU}-R_{\astrosun})}}{r_{AU}}\frac{\vec{r}_{AU}}{r_{AU}}\>,$
(96)

where $\vec{r}_{AU}$ is the vector pointing from Sun to Earth, $\vec{a}_{G}$
is the acceleration due to the Sun’s gravitational field and
$\vec{a}_{\phi{\earth}}$ the acceleration due to the dilaton field of the Sun
and given by Eq. (74), which holds for a test particle in an outer field
without screening. The screening of the Earth is taken into account by the
second charge factor $\mathfrak{Q}_{\earth}$. Likewise, the acceleration of
the Moon is given by

$\displaystyle\vec{a}_{\leftmoon}$
$\displaystyle=\vec{a}_{G}+\vec{a}_{\phi{\leftmoon}}$
$\displaystyle\simeq\vec{a}_{G}-\mathfrak{Q}_{\astrosun}\mathfrak{Q}_{\leftmoon}\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{\astrosun}^{2}R_{\astrosun}^{3}}{3}\,\phi_{V}\left(\phi_{V}-\phi_{\astrosun}\right)$
$\displaystyle\quad\times\frac{e^{-\mu_{V}(r_{AU}-R_{\astrosun})}}{r_{AU}}\frac{\vec{r}_{AU}}{r_{AU}}\>,$
(97)

where $\vec{r}_{AU}$ is again the vector pointing from Sun to Earth, which is
approximately the vector pointing from Sun to Moon. Then, $\vec{a}_{G}$ is the
acceleration due to the Sun’s gravitational field and taken equal for Earth
and Moon, such that any difference in acceleration between Earth and Moon,
viz. an equivalence principle violation, is attributed to the dilaton field.
For the equivalence principle violation quantified by the Eötvos parameter

$\displaystyle\eta_{\text{em}}=2\frac{|\vec{a}_{\earth}-\vec{a}_{\leftmoon}|}{|\vec{a}_{\earth}+\vec{a}_{\leftmoon}|}\simeq\frac{|\vec{a}_{\phi{\earth}}-\vec{a}_{\phi{\leftmoon}}|}{|\vec{a}_{G}|}\>,$
(98)

we find in the case of dilatons

$\displaystyle\eta_{\text{em}}$
$\displaystyle\simeq\frac{\mathfrak{Q}_{\astrosun}|\mathfrak{Q}_{\earth}-\mathfrak{Q}_{\leftmoon}|}{|\vec{a}_{G}|}\,\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{\astrosun}^{2}R_{\astrosun}^{3}}{3}\,\phi_{V}\left(\phi_{V}-\phi_{\astrosun}\right)$
$\displaystyle\quad\times\frac{e^{-\mu_{V}(r_{AU}-R_{\astrosun})}}{r_{AU}}\>.$
(99)

Typically this is constrained at the $\eta_{\rm em}\leq 2\times 10^{-13}$
level by the LLR experiment Williams:2012nc .

###  IX.2 Constraints due to Deviations from the Inverse-Square Law

For the second LLR test we consider the precession of the lunar perigee caused
by fifth forces. The corresponding Eq. (124) is derived in Appendix A and
reads

$\displaystyle\frac{\delta\Omega}{\Omega}\simeq-\frac{R^{2}}{GM_{\earth}}\left(\delta
f(R)+\frac{R}{2}\,\delta f^{\prime}(R)\right).$ (100)

It is derived in Appendix A. Here, $\delta f(R)$ is the acceleration caused by
dilatons at the maximum Earth-Moon separation $R$, and given by 74

$\displaystyle\delta f(R)$
$\displaystyle\simeq\mathfrak{Q}_{\earth}\mathfrak{Q}_{\leftmoon}\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{\earth}^{2}R_{\earth}^{3}}{3}\,\phi_{V}\left(\phi_{V}-\phi_{\earth}\right)$
$\displaystyle\quad\times\frac{e^{-\mu_{V}(R-R_{\earth})}}{R}\>.$ (101)

Using Eq. (IX.2) in Eq. (100) leads to the central relation

$\displaystyle\frac{\delta\Omega}{\Omega}$
$\displaystyle\simeq-\mathfrak{Q}_{\earth}\mathfrak{Q}_{\leftmoon}\frac{A_{2}}{m_{\text{Pl}}^{2}}\frac{\mu_{V}\mu_{\earth}^{2}R_{\earth}^{3}}{3}\frac{R}{GM_{\earth}}\,\phi_{V}\left(\phi_{V}-\phi_{\earth}\right)$
$\displaystyle\quad\times\left(1-\frac{\mu_{V}R}{2}\right)e^{-\mu_{V}(R-R_{\earth})}\>,$
(102)

to be compared with experimental results of order $6\times 10^{-12}$.

The results of the numerical analysis for both tests are reproduced in the
accompanying paper Pitschmann:2018aa .

##  X Conclusion

We have derived approximate analytical solutions to the dilaton field theory
in the presence of a one or two mirror system as well as for a sphere. The
1-dimensional equations of motion have been integrated in each case. The
analytical results obtained herein provide the necessary input for the
numerical study carried out in parallel to this work Pitschmann:2018aa . The
latter provides bounds on dilatons by using results from three ”experiments”.
First, from qBOUNCE we obtain results by employing bouncing ultracold
neutrons, second, Cannex provides bounds by measuring induced pressures
between parallel plates and third, from Lunar Laser Ranging we obtain
additional bounds in the astrophysical regime.

###### Acknowledgements.

We thank Hartmut Abele and René Sedmik for fruitful discussions. P.B.
acknowledges support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska -Curie grant agreement No
860881-HIDDeN. This work was supported by the Austrian Science Fund (FWF): P
34240-N.

##  Appendix A Precession of the Lunar Perigee induced by Fifth Forces

In this Appendix we derive the precession of the lunar perigee induced by an
arbitrary ”fifth force”. Historically, the following result is essentially due
to Isaac Newton (newton1999principia , Propositions XLIII - XLV). For a modern
derivation see chandrasekhar1995newton . In this Appendix, any ”force” is
understood as the absolute value of the centripetal force per mass

$\displaystyle f(r)=-\frac{\vec{r}}{r}\cdot\frac{d\vec{v}}{dt}\>.$ (103)

We recall the Binet equation governing $u=1/r=g(\varphi)$ (see e.g.
chandrasekhar1995newton )

$\displaystyle f(r)$
$\displaystyle=h^{2}u^{2}\frac{d^{2}u}{d\varphi^{2}}+h^{2}u^{3}$
$\displaystyle=h^{2}g^{2}(\varphi)\frac{d^{2}g}{d\varphi^{2}}(\varphi)+h^{2}g^{3}(\varphi)\>,$
(104)

where $h$ is the absolute value of $\vec{h}=\vec{r}\times\vec{v}$.

First, we address the following question. Given that $u=1/r=g(\varphi)$ is the
inverse polar equation of an orbit described under the action of a centripetal
force $f(r)$ with a constant of areas $h$, what is the centripetal force
$\tilde{f}(r)$ under which a revolving orbit $u=1/r=g(\tilde{\varphi}/\alpha)$
with $\tilde{\varphi}=\alpha\varphi$ and a constant of areas $\tilde{h}=\alpha
h$ would be described, where $\alpha$ is some assigned constant? The polar
form of an ellipse with its center at one focus and $\varphi$ defined such
that $\varphi=0$ corresponds to the point nearest to this focus, is given by

$\displaystyle\frac{p}{r}=1+e\cos\varphi$ (105)

with eccentricity $e$ and semi-latus rectum $p$. Hence, we have for an ellipse

$\displaystyle f(r)=\frac{h^{2}}{p}\frac{1}{r^{2}}\>.$ (106)

We now require the centripetal force $\tilde{f}(r)$ under which the orbit

$\displaystyle r=g(\tilde{\varphi}/\alpha)$ (107)

will be described with a constant of areas

$\displaystyle\tilde{h}=\alpha h\>.$ (108)

We find

$\displaystyle\tilde{f}(r)$
$\displaystyle=\tilde{h}^{2}u^{2}\frac{d^{2}u}{d\tilde{\varphi}^{2}}+\tilde{h}^{2}u^{3}$
$\displaystyle=\tilde{h}^{2}g^{2}(\tilde{\varphi}/\alpha)\frac{d^{2}g}{d\tilde{\varphi}^{2}}(\tilde{\varphi}/\alpha)+\tilde{h}^{2}g^{3}(\tilde{\varphi}/\alpha)$
$\displaystyle=\alpha^{2}h^{2}g^{2}(\varphi)\frac{1}{\alpha^{2}}\frac{d^{2}g}{d\varphi^{2}}(\varphi)+\alpha^{2}h^{2}g^{3}(\varphi)$
$\displaystyle=h^{2}g^{2}(\varphi)\frac{d^{2}g}{d\varphi^{2}}(\varphi)+\alpha^{2}h^{2}g^{3}(\varphi)$
$\displaystyle=f(r)-h^{2}g^{3}(\varphi)+\alpha^{2}h^{2}g^{3}(\varphi)$
$\displaystyle=f(r)-h^{2}u^{3}+\alpha^{2}h^{2}u^{3}\>,$ (109)

and finally Newton’s theorem of revolving orbits whittaker1988treatise

$\displaystyle\tilde{f}(r)-f(r)=\frac{\tilde{h}^{2}-h^{2}}{r^{3}}\>.$ (110)

In order to relate fifth forces to the precession of the Lunar perigee, we
compare the elliptic orbit described under an inverse-square law of attraction
with a nearly circular orbit caused by an inverse-square law of attraction
together with a small fifth force. The elliptic orbit described by Eq. (105)
is not revolving, while an additional fifth force leads to a deformation of
the orbit and also lets the orbit revolve. Actually, the orbit of the Moon is
to first order an ellipse with manifold corrections at higher orders. These
higher orders also lead to corrections of the relation between fifth force and
precession of the Lunar perigee, which are sub-leading and hence will be
neglected.

For an elliptic orbit we have Eq. (106) and find for the force causing the
revolving orbit

$\displaystyle\tilde{f}(r)$
$\displaystyle=\frac{h^{2}}{p}\frac{1}{r^{2}}+\frac{\tilde{h}^{2}-h^{2}}{r^{3}}$
$\displaystyle=\frac{1}{p}\frac{1}{r^{3}}\left[rh^{2}+p\left(\tilde{h}^{2}-h^{2}\right)\right].$
(111)

For a nearly circular orbit, we may write

$\displaystyle r=R-\delta r\>,$ (112)

where $R$ is the maximum distance, which for an ellipse is $R=a(1+e)$. Hence,
we find

$\displaystyle\tilde{f}(r)=\frac{1}{p}\frac{1}{r^{3}}\left[\left(R-\delta
r\right)h^{2}+p\left(\tilde{h}^{2}-h^{2}\right)\right].$ (113)

The new force we describe in the form

$\displaystyle\tilde{f}(r)=\frac{1}{p}\frac{C(r)}{r^{3}}\>,$ (114)

where $C(r)$ is an arbitrary function. Since we consider nearly circular
orbits, we perform a Taylor expansion

$\displaystyle\tilde{f}(r)\simeq\frac{1}{p}\frac{C(R)-C^{\prime}(R)\delta
r}{r^{3}}\>,$ (115)

where $\displaystyle C^{\prime}(R)=\frac{dC}{dR}$. Eq. (113) leads to

$\displaystyle\left(R-\delta
r\right)h^{2}+p\left(\tilde{h}^{2}-h^{2}\right)=C(R)-C^{\prime}(R)\delta r\>.$
(116)

Comparing coefficients gives

$\displaystyle Rh^{2}+p\left(\tilde{h}^{2}-h^{2}\right)$
$\displaystyle=C(R)\>,$ $\displaystyle h^{2}$ $\displaystyle=C^{\prime}(R)\>.$
(117)

For nearly circular orbits we have to leading order $p=R$ and hence

$\displaystyle R\tilde{h}^{2}$ $\displaystyle=C(R)\>,$ $\displaystyle h^{2}$
$\displaystyle=C^{\prime}(R)\>.$ (118)

For $\alpha$ we find

$\displaystyle\alpha=\frac{\tilde{h}}{h}=\sqrt{\frac{1}{R}\frac{C(R)}{C^{\prime}(R)}}\>.$
(119)

Using Eq. (114) we can express $\alpha$ in terms of $\tilde{f}(r)$

$\displaystyle\alpha=\sqrt{\frac{\tilde{f}(R)}{3\tilde{f}(R)+R\tilde{f}^{\prime}(R)}}\>.$
(120)

Next, we consider the case of gravitational attraction due to the Earth and an
additional small fifth force $\delta f(r)$ acting in radial direction and
caused by the Earth. Hence, we have

$\displaystyle\tilde{f}(r)=\frac{GM_{{\earth}}}{r^{2}}+\delta f(r)\>.$ (121)

To leading order we obtain

$\displaystyle\alpha\simeq 1-\frac{R^{2}}{GM_{{\earth}}}\,\delta
f(R)-\frac{1}{2}\frac{R^{3}}{GM_{{\earth}}}\,\delta f^{\prime}(R)\>.$ (122)

The correction to the precession of the Lunar perigee is obviously given by

$\displaystyle\frac{\delta\Omega}{\Omega}=\alpha-1\>,$ (123)

so we can finally connect the fifth force with the precession of the Lunar
perigee

$\displaystyle\frac{\delta\Omega}{\Omega}\simeq-\frac{R^{2}}{GM_{{\earth}}}\left(\delta
f(R)+\frac{R}{2}\,\delta f^{\prime}(R)\right).$ (124)

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